Navier–Stokes Equations

Advances in Mechanics and Mathematics 34

Grzegorz Łukaszewicz Piotr Kalita

Navier– Stokes Equations

An Introduction with Applications

Advances in Mechanics and Mathematics Volume 34

Series Editors: David Y. Gao, Virginia Polytechnic Institute and State University Tudor Ratiu, École Polytechnique Fédérale Advisory Board: Ivar Ekeland, University of British Columbia Tim Healey, Cornell University Kumbakonam Rajagopal, Texas A&M University David J. Steigmann, University of California, Berkeley

More information about this series at http://www.springer.com/series/5613

Grzegorz Łukaszewicz • Piotr Kalita

Navier–Stokes Equations An Introduction with Applications

123

Grzegorz Łukaszewicz Faculty of Mathematics, Informatics, and Mechanics University of Warsaw Warszawa, Poland

Piotr Kalita Faculty of Mathematics and Computer Science Jagiellonian University in Krakow Krakow, Poland

ISSN 1571-8689 ISSN 1876-9896 (electronic) Advances in Mechanics and Mathematics ISBN 978-3-319-27758-5 ISBN 978-3-319-27760-8 (eBook) DOI 10.1007/978-3-319-27760-8 Library of Congress Control Number: 2015960205 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland

To Renata, Agata, and Jacek, with love (Grzegorz) To my beloved wife Kasia (Piotr)

Preface

Admittedly, as useful a matter as the motion of fluid and related sciences has always been an object of thought. Yet until this day neither our knowledge of pure mathematics nor our command of the mathematical principles of nature have a successful treatment. –Daniel Bernoulli

Incompressible Navier–Stokes equations describe the dynamic motion (flow) of incompressible fluid, the unknowns being the velocity and pressure as functions of location (space) and time variables. To solve those equations would mean to predict the behavior of the fluid under knowledge of its initial and boundary states. These equations are one of the most important models of mathematical physics. Although they have been a subject of vivid research for more than 150 years, there are still many open problems due to the nature of nonlinearity present in the equations. The nonlinear convective term present in the equations leads to phenomena such as eddy flows and turbulence. In particular the question of solution regularity for three-dimensional problem was appointed by Clay Mathematics Institute as one of the Millennium Problems, that is, the key problems in modern mathematics. This is, on one hand, due to the fact that the problem remains challenging and fascinating for mathematicians and, on the other hand, that the applications of the Navier– Stokes equations range from aerodynamics (drag and lift forces), through design of watercrafts and hydroelectric power plants, to the medical applications of the models of flow of blood in vessels. This book is aimed at a broad audience of people interested in the Navier–Stokes equations, from students to engineers and mathematicians involved in the research on the subject of these equations. It originated in part from a series of lectures of the first author given over the past 15 years at the Faculty of Mathematics, Informatics and Mechanics of the University of Warsaw; at summer schools at UNICAMP, Campinas, Brasil; and at Université Jean Monnet, Saint-Etienne, France. The lectures were based on the leading books on the then young theory of infinite dimensional dynamical systems, focused on mathematical physics, in particular, on Temam [220]; Chepyzhov and Vishik [61]; Doering and Gibbon [88]; Foia¸s, Manley, Rosa, and Temam [99]; and Robinson [197].

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The lectures at the Mathematics Faculty of the University of Warsaw were also attended by students and PhD students from the Faculty of Physics and Faculty of Geophysics, and it became clear that a routine mathematical lecture had to be extended to include additional aspects of hydrodynamics. Some students asked for “more physics and motivation” and “more real applications”; others were mainly interested in the mathematics of the Navier–Stokes equations, and yet others would like to see the Navier–Stokes equations in a more general context of evolution equations and to learn the theory of infinite dimensional dynamical systems on the research level. These several aspects of hydrodynamics well suited the tastes and interests of the lecturer, and also the second author was welcomed to join the project of the book at a later stage. In consequence, the audience of the book is threesome: Group I: Mathematicians, physicists, and engineers who want to learn about the Navier–Stokes equations and mathematical modeling of fluids Group II: University teachers who may teach a graduate or PhD course on fluid mechanics basing on this book or higher-level students who start research on the Navier–Stokes equations Group III: Researchers interested in the exchange of current knowledge on dynamical systems approach to the Navier–Stokes equations Although, in principle, all these three groups can find interest in all chapters of the book, Chaps. 2–7 are primarily targeted at Group I, Chaps. 3, 4, 7, 8, 11, and 12 aimed mainly at Group II, and Chaps. 7–16 for Group III. For a reader with reasonable background on calculus, functional analysis, and theory of weak solutions for PDEs, the whole book should be understandable. The book was planned to be a monograph which could also be used as a textbook to teach a course on fluid mechanics or the Navier–Stokes equations. Typical courses could be “Navier–Stokes equations”, “partial differential equations”, “fluid mechanics”, “infinite dimensional dynamics systems.” To this end many chapters of this book include exercises. Moreover, we did not restrain ourselves to include a number of figures to liven the text and make it more intuitive and less formal. We believe that the figures will be helpful. Special care was undertaken to keep the individual chapters self-contained as far as possible to allow the reader to read the book linearly (in linear portions). That demanded several small repetitions here and there. To understand the first chapters of this book, just the basic knowledge on calculus, that can be learned from any calculus textbook, should be enough. The book is planned to be self-contained, but, to understand its last chapters, some knowledge from a textbook like “Partial Differential Equations” by L.C. Evans (which contains all necessary knowledge on functional analysis and PDEs) would be helpful. Each chapter contains an introduction that explains in simple words the nature of presented results and a section on bibliographical notes that will place it in the context of past and current research.

Preface

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Several people greatly contributed, knowingly or not knowingly, to the creation of the book. Our thanks go to our colleagues and collaborators: Guy Bayada, Mahdi Boukrouche, Thomas Caraballo, José Langa, Pepe Real, James Robinson etc. The first author is grateful to Guy Bayada who introduced him to the problems of lubrication theory and flows in narrow films during his visits at INSA, Lyon, and to Mahdi Boukrouche with whom he collaborated for several years on this subject. Thomas Caraballo, José Langa, and Pepe Real introduced him to the subject of pullback attractors during his visit at the University of Seville. Thanks for the opportunity to give the summer courses in Campinas and Saint-Etienne go to Marco Rojas-Medar and Mahdi Boukrouche, respectively. Many thanks go also to Chunyou Sun, Meihua Yang, and Yongqin Xie for their kind invitation of the first author to give several lectures at the University of Lanzhou, then at Huazhong University of Science and Technology in Wuhan and at The University in Changsha, China, in June 2013. Inspirational discussions and exchange of ideas with these Chinese friends, including also Qingfeng Ma and Yuejuan Wang, contributed to the form of the last chapters of the book. The research group at Jagiellonian University with their leader, Stanisław Migórski, greatly motivated the authors as regards contact problems. The second author owes a lot to his colleagues and teachers from Jagiellonian University; he would like to express his thanks especially to Zdzisław Denkowski and Stanisław Migórski. He would also like to thank Robert Schaefer who first introduced to him the topics of fluid mechanics. He is also grateful for inspiring discussions in the field of contact mechanics to his collaborators from the University of Perpignan, Mircea Sofonea and Mikaël Barboteu. We would like to thank Wojciech Pociecha for his help with the preparation of the figures. The work was in part supported by the National Science Center of Poland under the Maestro Advanced Project no. DEC-2012/06/A/ST1/00262. Finally, we express our gratitude to the AMMA Series editor, David Y. Gao, and to Marc Strauss and the editors of Springer Publishing House for their care and encouragement during the preparation of the book. Warszawa, Poland Kraków, Poland October 2015

Grzegorz Łukaszewicz Piotr Kalita

Contents

1

Introduction and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Equations of Classical Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Derivation of the Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Stress Tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Vorticity Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Similarity of Flows and Nondimensional Variables . . . . . . . . . . . . . . . . 2.8 Examples of Simple Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 20 22 23 24 26 28 31 36

3

Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Theorems from Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Sobolev Spaces and Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Some Embedding Theorems and Inequalities. . . . . . . . . . . . . . . . . . . . . . . 3.4 Sobolev Spaces of Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Evolution Spaces and Their Useful Properties . . . . . . . . . . . . . . . . . . . . . . 3.6 Gronwall Type Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Clarke Subdifferential and Its Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Nemytskii Operator for Multifunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Clarke Subdifferential: Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 44 48 54 61 67 70 74 77 81

4

Stationary Solutions of the Navier–Stokes Equations . . . . . . . . . . . . . . . . . . 4.1 Basic Stationary Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 The Stokes Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 The Nonlinear Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Other Topological Methods to Deal with the Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 84 86 90 93 xi

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5

Stationary Solutions of the Navier–Stokes Equations with Friction . . 95 5.1 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.2 Friction Operator and Its Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.3 Weak Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.4 Existence of Weak Solutions for the Case of Linear Growth Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.5 Existence of Weak Solutions for the Case of Power Growth Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.6 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6

Stationary Flows in Narrow Films and the Reynolds Equation . . . . . . . 6.1 Classical Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Weak Formulation and Main Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Scaling and Uniform Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Limit Variational Inequality, Strong Convergence, and the Limit Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Remarks on Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Strong Convergence of Velocities and the Limit Equation . . . . . . . . . 6.7 Reynolds Equation and the Limit Boundary Conditions . . . . . . . . . . . 6.8 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

111 111 114 121 125 128 134 137 141 142

Autonomous Two-Dimensional Navier–Stokes Equations . . . . . . . . . . . . . 7.1 Navier–Stokes Equations with Periodic Boundary Conditions . . . . 7.2 Existence of the Global Attractor: Case of Periodic Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Convergence to the Stationary Solution: The Simplest Case . . . . . . . 7.4 Convergence to the Stationary Solution for Large Forces . . . . . . . . . . 7.5 Average Transfer of Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143 143

8

Invariant Measures and Statistical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Existence of Invariant Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Stationary Statistical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169 169 176 181

9

Global Attractors and a Lubrication Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Fractal Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Abstract Theorem on Finite Dimensionality and an Algorithm . . . . 9.3 An Application to a Shear Flow in Lubrication Theory . . . . . . . . . . . . 9.3.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Energy Dissipation Rate Estimate . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 A Version of the Lieb–Thirring Inequality . . . . . . . . . . . . . . . . 9.3.4 Dimension Estimate of the Global Attractor . . . . . . . . . . . . . . 9.4 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183 183 185 193 193 196 200 201 204

151 157 159 163 166

Contents

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11

12

Exponential Attractors in Contact Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Exponential Attractors and Fractal Dimension . . . . . . . . . . . . . . . . . . . . . 10.2 Planar Shear Flows with the Tresca Friction Condition . . . . . . . . . . . . 10.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Existence and Uniqueness of a Global in Time Solution . 10.2.3 Existence of Finite Dimensional Global Attractor . . . . . . . . 10.2.4 Existence of an Exponential Attractor . . . . . . . . . . . . . . . . . . . . . 10.3 Planar Shear Flows with Generalized Tresca Type Friction Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Classical Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . 10.3.2 Weak Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Existence and Properties of Solutions . . . . . . . . . . . . . . . . . . . . . 10.3.4 Existence of Finite Dimensional Global Attractor . . . . . . . . 10.3.5 Existence of an Exponential Attractor . . . . . . . . . . . . . . . . . . . . . 10.4 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-autonomous Navier–Stokes Equations and Pullback Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Determining Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Determining Nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Pullback Attractors for Asymptotically Compact Non-autonomous Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Application to Two-Dimensional Navier–Stokes Equations in Unbounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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207 207 211 211 218 222 229 231 231 233 236 241 246 248 251 251 256 260 268 274

Pullback Attractors and Statistical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Pullback Attractors and Two-Dimensional Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Construction of the Family of Probability Measures . . . . . . . . . . . . . . . 12.3 Liouville and Energy Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Time-Dependent and Stationary Statistical Solutions . . . . . . . . . . . . . . 12.5 The Case of an Unbounded Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

277 277 281 285 288 291 295

13

Pullback Attractors and Shear Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Existence and Uniqueness of Global in Time Solutions. . . . . . . . . . . . 13.4 Existence of the Pullback Attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Fractal Dimension of the Pullback Attractor . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

297 297 298 301 305 310 316

14

Trajectory Attractors and Feedback Boundary Control in Contact Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 14.1 Setting of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 14.2 Weak Formulation of the Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

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14.3 14.4 14.5 15

16

Existence of Global in Time Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 Existence of Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

Evolutionary Systems and the Navier–Stokes Equations . . . . . . . . . . . . . . 15.1 Evolutionary Systems and Their Attractors . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Three-Dimensional Navier–Stokes Problem with Multivalued Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Existence of Leray–Hopf Weak Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Existence and Invariance of Weak Global Attractor, and Weak Tracking Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

337 337

Attractors for Multivalued Processes in Contact Problems. . . . . . . . . . . . 16.1 Abstract Theory of Pullback D-Attractors for Multivalued Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Application to a Contact Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

359

340 342 351 356

359 366 376

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

1

Introduction and Summary

When you put together the science of movements of water, remember to put beneath each proposition its applications, so that such science may not be without uses. – Leonardo da Vinci

This chapter provides, for the convenience of the reader, an overview of the whole book, first of its structure and then of the content of the individual chapters. The outline of the book structure is as follows. Chapter 2 shows the derivation of the Navier–Stokes equations from the principles of physics and discusses their physical and mathematical properties and examples of the solutions for some particular cases, without going into complicated mathematics. This part is aimed to fill in a gap between an engineer and mathematician and should be understood by anybody with basic knowledge of calculus no more complicated than the Stokes theorem. In Chap. 3, a necessary mathematical background including these parts of functional analysis and theory of Sobolev spaces which are needed to understand modern research on the Navier–Stokes equations is presented. Chapters 4–6 comprise three examples of stationary problems. Then we smoothly move to the research level part of the book (Chaps. 7–16) which presents the analysis from the point of view of global attractors of the asymptotic (in time) behavior of the velocities being the solutions of the Navier– Stokes system. Roughly speaking we endeavor to show how the modern theory of global attractors can be used to construct the mathematical objects that enclose the seemingly chaotic and unordered eddy and turbulent flows. We tame these flows by showing their fine properties like the finite dimensionality of global attractors, which means that the description of unrestful and turbulent states can be done by finite number of parameters or existence of invariant measures which means that the flow becomes, in statistical sense, stationary. We deal with non-autonomous problems using the recent and elegant theory of so-called pullback attractors that allows to cope with flows with changing in time sources, sinks, or boundary data. We also solve problems with multivalued boundary conditions that allow to model various contact conditions between the fluid and enclosing object, such as the stick/slip frictional boundary behavior.

© Springer International Publishing Switzerland 2016 G. Łukaszewicz, P. Kalita, Navier–Stokes Equations, Advances in Mechanics and Mathematics 34, DOI 10.1007/978-3-319-27760-8_1

1

2

1 Introduction and Summary

The analysis is primarily done for the two-dimensional Navier–Stokes equations, where, as a model, the problem from lubrication theory, that can be reduced to two dimensions is used. The study with various types of boundary conditions, including multivalued ones, is presented. Some of the results presented in this part are based on the previous and already published work of the authors, but some results, that are the subjects of current research, are yet unpublished elsewhere. Below we present the content of the book in some more detail. In Chap. 2 we give an overview of the equations of classical hydrodynamics. We provide their derivation, discuss the associated physical quantities, comment on the constitutive laws, stress tensor, and thermodynamics, finally we present some elementary properties of the derived system and also some cases where it is possible to calculate the exact solutions of the following system of the incompressible Navier–Stokes equations, @u 1 u C .u r/u C rp D f ; @t div u D 0; which are the main subject of the book. In Chap. 3 we introduce the basic preliminary mathematical tools to study the Navier–Stokes equations, including results from linear and nonlinear functional analysis (e.g., the Lax–Milgram lemma, fixed point theorems) as well as the theory of function spaces (e.g., compactness theorems). We present, in particular, some of the most frequently used in the sequel embedding theorems and inequalities. We discuss the versions of the Gronwall lemma used in the sequel, and provide some necessary background in the theory of Clarke subdifferential and differential inclusions. In Chaps. 4–6 we consider stationary problems. Chapter 4 is devoted to the stationary Navier–Stokes equations in a bounded three-dimensional domain Q, u C .u r/u C rp D f

in Q;

divu D 0 in Q; with one of the boundary conditions: 1. Q D Œ0; L3 in R3 and we assume periodic boundary conditions, or 2. Q is a bounded domain in R3 , with smooth boundary, and we assume the homogeneous boundary condition u D 0 on @Q. This basic problem serves as an introduction to the mathematical theory of the Navier–Stokes equations. We introduce the suitable function spaces in which we (usually) seek solutions of the stationary problem, then we present the weak formulation. It allows us to use the theories of linear and nonlinear functional analysis (Lax–Milgram lemma and fixed point theorems, respectively) to prove the existence of solutions.


3

To show typical methods used when dealing with nonlinear problems, we present a number of proofs based on various linearizations and fixed point theorems in standard function spaces. The solutions, due to the nonlinearity of the Navier–Stokes equations are not in general unique, however, under some restriction on the mass force and viscosity coefficient (quite intuitive from the physical point of view), one can prove their uniqueness. In Chap. 5 we consider the stationary Navier–Stokes equations with friction in the three-dimensional bounded domain ˝. The domain boundary @˝ is divided into two parts, namely the boundary D on which we assume the homogeneous Dirichlet boundary condition and the contact boundary C on which we decompose the velocity into the normal and tangent directions. In the normal direction we assume u n D 0, i.e., there is no leak through the boundary, while in the tangent direction we set T 2 h.x; u /, the multivalued relation between the tangent stress and tangent velocity. This relation is the general form of the friction law on the contact boundary. The existence of solution is shown by the Kakutani–Fan– Glicksberg fixed point theorem and some cut-off techniques. In Chap. 6 we consider a typical problem for the hydrodynamic equations coming from the lubrication theory. In this theory the domain of the flow is usually very thin and the engineers are interested in the distribution of the pressure therein. Because of the thinness of the domain, in practice one assumes that the pressure would depend only on two and not three independent variables. The pressure distribution is governed then by the Reynolds equation, which depends on the original boundary data. Our aim is to obtain the Reynolds equation starting from the stationary Stokes equations, considered in the three-dimensional domains ˝ " , " > 0, u C rp D f div u D 0

in

˝ ";

in ˝ " ;

with a Fourier boundary condition on the top F" and Tresca boundary conditions on the bottom part of the boundary C , respectively. We show how to pass, in a precise mathematical way, as " ! 0, from the three-dimensional Stokes equations to a two-dimensional Reynolds equation for the pressure distribution (see Fig. 1.1). The passage from the three-dimensional problem to a two-dimensional one depends on several factors and additional scaling assumptions. The existence of solutions of the limit equations follows from the existence of solutions of the original three-dimensional problem. Finally, we show the uniqueness of the limit solution. In Chaps. 7–10 we consider the nonstationary autonomous Navier–Stokes equations ut u C .u r/u C rp D f ;

(1.1)

div u D 0;

(1.2)

in two-dimensional domains. In these chapters the solutions are global in time and unique.

4


Fig. 1.1 Schematic view of the problem considered in Chap. 6. Incompressible static Stokes equation is solved on the three-dimensional domain ˝ " . The domain thickness is given by "h, where h is a function of the point in the two-dimensional domain !. As " ! 0, Reynolds equation on ! is recovered

Chapter 7 is a general introduction to evolutionary two-dimensional Navier– Stokes equations. We prove some basic properties of solutions assuming that the external forces do not depend on time and that the domain of the flow is bounded. The boundary conditions are either periodic or homogeneous Dirichlet ones. In this chapter we introduce the notion of the global attractor, one of the main objects to study also in Chaps. 8–10. In Chap. 8 we prove the existence of invariant measures associated with twodimensional autonomous Navier–Stokes equations. The invariant measures are supported on the global attractor. Then we introduce the notion of a stationary statistical solution and prove that every invariant measure is also such a solution. Existence of the invariant measures (stationary statistical solutions) supported on the global attractor reveals the statistical properties of the potentially chaotic fluid flow after a long time of evolution when the external forces do not depend on time. The non-autonomous case is considered in Chap. 12. In Chaps. 9–10 we consider system (1.1) and (1.2) in the domain ˝ depicted in Fig. 1.2, with homogeneous condition u D 0 on D , periodic condition u.0; x2 / D u.L; x2 / on L , and several contact boundary conditions on C . The motivation for such problem setup comes again from problems in contact mechanics, the theory of lubrication and shear flows in narrow films. One may look at the domain ˝ as a rectification of the ring-like domain considered in the theory of lubrication, where it represents a cross section of an infinite journal bearing. The problem reduces to describing a flow between two cylinders. The outer cylinder is at rest and the inner cylinder rotates providing a driving force to the fluid (lubricant). Since the cylinders are infinitely long it can be assumed in the first approximation that the flow is two-dimensional. Described domain geometry is schematically presented in Fig. 1.3. The boundary conditions on C include the following ones. In Chap. 9 we pose u D Ue1 ;

U 2 R;

U>0

on

C ;


5

Fig. 1.2 Schematic view of the flow domain and its boundaries in Chaps. 9 and 10

Fig. 1.3 Three-dimensional infinite ring-like domain and its rectification considered in Chaps. 9 and 10

by which we mean that the boundary C is moving with a constant velocity U0 e1 D .U0 ; 0/ and the velocity of the fluid at the boundary equals the velocity of the boundary. We prove the existence of a global attractor and estimate from above its fractal dimension in terms of given data and geometry of the domain of the flow. In Chap. 10 we consider two problems. We assume that there is no flux across C so that the normal component of the velocity on C satisfies unD0

on C ;

and that the tangential component of the velocity u on C is unknown and satisfies the Tresca friction law with a constant and positive maximal friction coefficient k. This means that

6


9 > > > > > =

jT .u; p/j k jT .u; p/j < k ) u D U0 e1 jT .u; p/j D k ) 9 0 such that u D U0 e1 T .u; p/

> > > > > ;

on

C ;

(1.3) where T is the tangential component of the stress tensor on C and U0 e1 D .U0 ; 0/, U0 2 R, is the velocity of the lower surface producing the driving force of the flow. In the second problem the boundary C is also assumed to be moving with the constant velocity U0 e1 D .U0 ; 0/ which, together with the mass force, produces the driving force of the flow. The friction coefficient k is assumed to be related to the slip rate through the relation k D k.ju U0 j/, where k W RC ! RC . If there is no slip between the fluid and the boundary then the friction is bounded by the threshold k.0/ u D U0 ) jT j k.0/

on

C ;

(1.4)

while if there is a slip, then the friction force density (equal to tangential stress) is given by the expression u ¤ U0 ) T D k.ju U0 j/

u U0 ju U0 j

on

C :

(1.5)

Note that (1.4) and (1.5) generalize the Tresca law (1.3) where k was assumed to be a constant. Here k depends of the slip rate, this dependence represents the fact that the kinetic friction is less than the static one, which holds if k is a decreasing function. We prove that for both problems above there exist exponential attractors, in particular the global attractors of finite fractal dimension. In Chaps. 11–13 we consider the time asymptotics of solutions of the two-dimensional Navier–Stokes equations. First, in Chap. 11 we prove two properties of the equations in a bounded domain, concerning the existence of determining modes and nodes. Then we study the equations in an unbounded domain, in the framework of the theory of infinite dimensional non-autonomous dynamical systems, introducing the notion of the pullback attractor. Chapter 12 presents a construction of invariant measures and statistical solutions for the non-autonomous Navier–Stokes equations in bounded and some unbounded domains in R2 . More precisely, we construct the family of probability measures f t gt2R and prove the relations t .E/ D .U.t; /1 E/ for t; 2 R, t and Borel sets E in H. The support of each measure t is included in the section A.t/ of the pullback attractor. We prove also the Liouville and energy equations. Finally, we consider statistical solutions of the Navier–Stokes equations supported on the pullback attractor.


7

In Chap. 13 we consider the problem of existence and finite dimensionality of the pullback attractor for a class of two-dimensional turbulent boundary driven flows. We generalize here the results from Chap. 9 to the non-autonomous problem. The new element in our study with respect to that in Chap. 9 is the allowance of the velocity of C to depend on time, i.e., u D U.t/e1 ;

U.t/ 2 R on

C :

Our aim is to study the time asymptotics of solutions in the frame of the dynamical systems theory. We prove the existence of the pullback attractor and estimate its fractal dimension. We shall apply the results from Chap. 11, reformulated here in the language of evolutionary processes. Chapters 14–16 are devoted to global in time solutions of the Navier–Stokes equations which are not necessary unique. We introduce theories of global attractors for multivalued semiflows and multivalued processes to include this situation. We study further examples of contact problems in both autonomous and nonautonomous cases. In Chap. 14 we consider two-dimensional nonstationary Navier–Stokes shear flows in the domain ˝ as in Fig. 1.2, with nonmonotone and multivalued boundary conditions on C . Namely, we assume the following subdifferential boundary condition pQ .x; t/ 2 @j.un .x; t//

on

C ;

where pQ D p C 12 juj2 is the Bernoulli (total) pressure, un D u n, j W R ! R is a given locally Lipschitz superpotential, and @j is a Clarke subdifferential of j. Our considerations are motivated here by feedback control problems for fluid flows in domains with semipermeable walls and membranes and by the theory of lubrication. We prove the existence of global in time solutions of the considered problem which is governed by a partial differential inclusion, and then we prove the existence of a trajectory attractor and a weak global attractor for the associated multivalued semiflow. In Chap. 15 we study the three-dimensional problem in a bounded domain ˝. The problem domain is the three-dimensional counterpart of the one presented in Fig. 1.2. The boundary of ˝ is divided into three parts: the lateral one L on which we assume the periodic boundary conditions, the homogeneous Dirichlet one and, finally, the contact one C on which we consider a general form of multivalued frictional type boundary conditions T 2 g.u /. We prove the existence of the Leray–Hopf weak solutions and, using the framework of evolutionary systems, existence of the weak global attractor. Finally, in Chap. 16 we consider further non-autonomous and multivalued evolution problems, this time in the frame of the theory of pullback attractors for multivalued processes. First we prove an abstract theorem on the existence of pullback D-attractor and then apply it to study a two-dimensional incompressible

8


Navier–Stokes flow with a general form of multivalued frictional contact conditions on C . We assume that there is no flux across C and hence we have un .t/ D 0

on

C ;

and that the tangential component of the velocity u on C is in the following relation with the tangential stresses T , T .t/ 2 @j.x; t; u .t//

on

C :

In above formula j W C R R ! R is a potential which is locally Lipschitz and not necessarily convex with respect to the last variable, and @ is the subdifferential in the sense of Clarke taken with respect to the last variable u . The tangent conditions on C in Chaps. 15 and 16 represent the frictional contact between the fluid and the wall, where the friction force depends in a nonmonotone and even discontinuous way on the slip rate, and are a generalization of the conditions considered in Chap. 10. For this case we prove the existence of the attractor. Most chapters are devoted to two-dimensional problems. Three-dimensional problems are considered only in Chaps. 2, 4, 5, 6, and 15. One reason for that is associated with the character of the Navier–Stokes equations, namely the fact that in the two-dimensional problems it is relatively easy to prove the uniqueness of the solutions which allows us to use the well-developed theory of infinite dimensional dynamical systems for semigroup and processes, while the uniqueness of the threedimensional Navier–Stokes equations is in general an open question. We also consider the two- and three-dimensional problems without assuming the solution uniqueness in the framework of (more recent) theories of trajectory attractors, multivalued semiflows, evolutionary systems, and multivalued processes. The other reason to focus on the two-dimensional flows concern the simplicity. Our aim was to test first the more elementary two-dimensional models of some real engineering problems. The word “test” here means not only checking the well posedness of a particular problem. In Chap. 9 we estimate the attractor dimension and show how it depends on the shape of the domain (cf. [24, 26], where the upper bounds of the attractor dimension depend also on the geometry of D ). Assume that the answer to the question on the dependence of the attractor dimension on the geometry of the boundary is such that in the two-dimensional case the estimate from above of the attractor dimension is independent of geometry (for example, on the roughness of the boundary represented by the oscillations of the function h D h.x1 / describing D ). Such a result would be contradictory to our intuition, provided the intuitive hypotheses attractor dimension level of chaos in the flow geometry of the flow domain

where “” means “is related to,” are justified. Such a contradiction with our intuition could be resolved in the following ways:


9

1. there is no such contradiction in the “real” three-dimensional case, it appears only in the two-dimensional case (but where lies the difference?), 2. the attractor dimension does not represent the level of chaos in the fluid flow described by the (good) model of the Navier–Stokes equations, 3. the Navier–Stokes equations model is not good enough to give the right answer to the problems of chaotic movement of the classical fluids. The close correspondence between the level of chaos in the fluid flow and the geometry of the domain is evident as a physical phenomenon (recall observing a flow of water in the river). To confirm the agreement of the results provided by the modeling with our physical intuition or else to confront the above potential possibilities motivated us to study the problems of the existence and properties of the attractor. There are still many interesting and important problems close to these considered in the book and we were able to touch only a few ones. One example is to further study the relations between the (type of) boundary conditions and the attractor dimension. Finally, we remark that this book is devoted to incompressible flows, for the mathematical treatise of compressible ones see, e.g., [157, 187].

2

Equations of Classical Hydrodynamics

The neglected borderline between two branches of knowledge is often that which best repays cultivation, or, to use a metaphor of Maxwell’s, the greatest benefits may be derived from a cross-fertilization of the sciences. – John William Strutt, 3rd Baron Rayleigh

In this chapter we give an overview of the equations of classical hydrodynamics. We provide their derivation, comment on the stress tensor, and thermodynamics, finally we present some elementary properties and also some exact solutions of the Navier–Stokes equations.

2.1 Derivation of the Equations of Motion Fluid flow may be represented mathematically as a continuous transformation of three-dimensional Euclidean space into itself. The transformation is parametrized by a real parameter t representing time. Let us introduce a fixed rectangular coordinate system .x1 ; x2 ; x3 /. We refer to the coordinate triple .x1 ; x2 ; x3 / as the position and denote it by x. Now consider a particle P moving with the fluid, and suppose that at time t D 0 it occupies a position X D .X1 ; X2 ; X3 / and that at some other time t, 1 < t < C1, it has moved to a position x D .x1 ; x2 ; x3 /. Then x is determined as a function of X and t x D x.X; t/

or

xi D xi .X; t/ :

(2.1)

If X is fixed and t varies, Eq. (2.1) specifies the path of the particle initially at X. On the other hand, for fixed t, (2.1) determines a transformation of a region initially occupied by the fluid into its position at time t.


11

12

2 Equations of Classical Hydrodynamics

We assume that the transformation (2.1) is continuous and invertible, that is, there exists its inverse X D X.x; t/;

.or

Xi D Xi .x; t// :

Also, to be able to differentiate, we assume that the functions xi and Xi are sufficiently smooth. From the condition that the transformation (2.1) possess a differentiable inverse it follows that its Jacobian @xi J D J.X; t/ D det @Xj satisfies 0 < J < 1:

(2.2)

The initial coordinates X of the particle will be referred to as the material coordinates of the particle. The spatial coordinates x may be referred to as its position, or place. The representation of fluid motion as a point transformation violates the concept of the kinetic theory of fluids, as in this theory the particles are molecules, and they are in random motion. In the theory of continuum mechanics the state of motion at a given point x and at a given time t is described by a number of functions such as D .x; t/, u D u.x; t/, D .x; t/ representing density, velocity, temperature, and other hydrodynamical variables. Due to the transformation (2.1), each such variable f can also be expressed in terms of material coordinates f .x; t/ D f .x.X; t/; t/ D F.X; t/ :

(2.3)

The velocity u at time t of a particle initially at X is given, by definition, as u.x; t/ D U.X; t/ D

d x.X; t/ ; dt

.x D x.X; t// :

(2.4)

Above, X is treated as a parameter representing a given fixed particle, and this is the reason that we use the ordinary derivative in (2.4). Having the velocity field u.x; t/, we can (in principle) determine the transformation (2.1), solving the ordinary differential equation d x.X; t/ D u.x.X; t/; t/; dt with x.X; 0/ D X, where X is a parameter. We shall always write d F.X; t/ dt

and

@ f .x; t/ ; @t

2.1 Derivation of the Equations of Motion

13

where F and f are related by (2.3). We have thus dxi d d @f @f .x.X; t/; t/ F.X; t/ D f .x.X; t/; t/ D C .x.X; t/; t/ ; dt dt @xi dt @t so that by (2.4) we obtain a general formula D d F.X; t/ D f .x; t/ ; dt Dt where

D f .x; t/ Dt

@f .x; t/ @t

(2.5)

C u.x; t/ rf .x; t/ is called the material derivative of f .

Transport Theorem Let ˝.t/ denote an arbitrary volume that is moving with the fluid and let f .x; t/ be a scalar or vector function of position and time. The transport theorem states that Z d f .x; t/ dx (2.6) dt ˝.t/

Z D ˝.t/

@f .x; t/ C u.x; t/ rf .x; t/ C f .x; t/ div u.x; t/ dx : @t

For the proof consider the transformation x W ˝.0/ ! ˝.t/;

x D x.X; t/ ;

as in (2.1). Then Z ˝.t/

f .x; t/ dx Z

D ˝.0/

Z f .x.X; t/; t/J.X; t/ dX D

˝.0/

F.X; t/J.X; t/ dX ;

so that d dt

Z ˝.t/

f .x; t/dx D

d dt

Z F.X; t/J.X; t/ dX ˝.0/

Z D ˝.0/

d d F.X; t/J.X; t/ C F.X; t/ J.X; t/ dX : dt dt

(2.7)

14


By (2.5) we have Z

d F.X; t/J.X; t/ dX dt

˝.0/

Z

@f .x.X; t/; t/ C u.x.X; t/; t/ rf .x.X; t/; t/ J.X; t/ dX @t

@f .x; t/ C u.x; t/ rf .x; t/ dx : @t

D ˝.0/

Z D ˝.t/

To prove (2.6) it remains to prove the Euler formula d J.X; t/ D div u.x.X; t/; t/J.X; t/ ; dt

(2.8)

the proof of which we leave to the reader as an exercise. The fluid is called incompressible if for any domain ˝.0/ and any t, volume .˝.t// D volume .˝.0// : From (2.7) with f .x; t/ 1 we have d d volume .˝.t// D dt dt

Z

Z dx D ˝.t/

˝.0/

d J.X; t/dX ; dt

hence by (2.8), (2.2), and the arbitrariness of choice of the domain ˝.t/ via ˝.0/ a necessary and sufficient condition for the fluid to be incompressible is div u.x; t/ D 0 : Exercise 2.1. Prove that the transport theorem can be written in the form d dt

Z

Z ˝.t/

f .x; t/ dx D

˝.t/

@f .x; t/ dx C @t

Z @˝.t/

f .x; t/u.x; t/ n.x; t/ dS ;

where n.x; t/ is the outward unit normal to @˝.t/ at x 2 @˝.t/. Equation of Continuity Let D .x; t/ be the mass per unit volume of a fluid at point x and time t. Then the mass of any finite volume ˝ is Z mD ˝

.x; t/ dx :


15

The principle of conservation of mass says that the mass of a fluid in a material volume ˝ does not change as ˝ moves with the fluid; that is, d dt

Z ˝.t/

.x; t/ dx D 0 :

From the transport theorem (2.6) it follows that

Z ˝.t/

@ C div .u/ dx D 0 ; @t

whence @ C div .u/ D 0 : @t

(2.9)

Sometimes the principle of conservation of mass is expressed as follows. Let ˝ be a fixed volume. Then Z Z d .x; t/ dx D u n dS ; (2.10) dt ˝ @˝ that is, the rate of change of mass in a fixed volume ˝ is equal to the mass flux through its surface. We notice also the general formula d dt

Z

Z ˝.t/

f dx D

˝.t/

D f dx : Dt

(2.11)

Exercise 2.2. Derive (2.9) from (2.10). Exercise 2.3 (Cf. [212]). Show that in material coordinates the equation of continuity is d f.X; t/J.X; t/g D 0 ; dt or .X; t/J.X; t/ D .X; 0/ : Exercise 2.4 (Cf. [5]). Show that if 0 .X/ is the distribution of density of the fluid at time t D 0 and r.div u/ D 0, then .x; t/ D 0 .X.x; t// exp

Z

t 0

div u.x; t/ dt :

16


Exercise 2.5. Find .x; t/ for the motion ui D

xi 1 C ai t

.a1 D 2 ; a2 D 1 ; a3 D 0/;

if 0 .X/ is the distribution of density of the fluid at time t D 0. Exercise 2.6. Prove (2.11). Principle of Conservation of Linear Momentum We assume that the forces acting on an element of a continuous medium are of two kinds. External, or body, forces, such as gravitation or electromagnetic forces, can be regarded as reaching into the medium and acting throughout the volume. If f represents such a force per unit mass, then it acts on an element ˝ as Z f dx : ˝

The internal, or contact, forces are to be regarded as acting on an element of volume ˝ through its bounding surface. Let n be the unit outward normal at a point of the surface @˝, and tn the force per unit area exerted there by the material volume outside @˝. Then the surface force exerted on the volume ˝ can be expressed by the integral Z @˝

tn dS :

The Cauchy principle says that tn depends at any given time only on the position and the orientation of the surface element dS; in other words, tn D tn .x; t; n/ : The principle of conservation of linear momentum says that the rate of change of linear momentum of a material volume equals the resultant force on the volume d dt

Z

Z ˝.t/

udx D

Z ˝.t/

f dx C

@˝.t/

tn dS ;

(2.12)

tn dS :

(2.13)

where f is assumed to be known. By (2.11), (2.12) yields Z ˝.t/

Du dx D Dt

Z

Z ˝.t/

f dx C

@˝.t/

From this equation we derive a very important fact, namely, that the vector tn (called normal stress) can be expressed as a linear function of n, in the form tn .x; t; n/ D n.x; t/T.x; t/ ;

(2.14)


17

where T D fTij g is a matrix called the stress tensor. This will allow us to pass from the integral form (2.13) of the equation of conservation of linear momentum to a differential one. Let l3 be the volume of ˝ D ˝.t/. Dividing both sides of (2.13) by l2 and letting the volume tend to zero we obtain Z 2 lim l tn dS D 0 ; (2.15) j˝j!0

@˝

that is, the stress forces are in local equilibrium. Let ˝ be a domain containing a fluid, and consider a regular tetrahedron with vertex at an arbitrary point x 2 @˝, and with three of its faces parallel to the coordinate planes. Let the slanted face have normal n D .n1 ; n2 ; n3 / and area ˙. The normals to the other faces are e1 , e2 , and e3 , and their areas are n1 ˙, n2 ˙, and n3 ˙ . Applying (2.15) to the family of tetrahedrons obtained by letting ˙ ! 0, we obtain t.n/ C n1 t.e1 / C n2 t.e2 / C n3 t.e3 / D 0 ;

(2.16)

where t.n/ D tn D tn .x; t; n/, t.h/ D th for h 2 fe1 ; e2 ; e3 g, and ni > 0. By a continuity argument, (2.16) holds for all ni 0, and then we prove easily that t.ei / D t.ei /, i D 1; 2; 3, and that it holds for all n. This means that t.n/ may be expressed as a linear function of n; that is, we can write it in the form (2.14). Thus, by (2.13) and by the Green theorem we obtain Z ˝.t/

Du dx D Dt

Z ˝.t/

.f C div T/ dx ;

whence, by the arbitrariness of the domain of integration,

Du D f C div T ; Dt

(2.17)

or 0

1 3 X @ @ @ ui C uj ui A D fi C Tji;j ; @t @xj jD1

i D 1; 2; 3 :

This is the general Cauchy equation of motion in differential form. Exercise 2.7. Give a physical interpretation of the components of the stress tensor. Notice that we have not specified T yet, that is, we have not made any assumptions concerning the nature of forces acting on surface elements. These forces depend on the kind of fluid, or, more generally, on the kind of medium under consideration.

18


In the simplest model the contact forces act perpendicularly to the surface elements. We have then t.n/ D p.x/n ; and call p the pressure. The minus sign is chosen so that when p > 0, the contact forces on a closed surface tend to compress the fluid inside; p represents the pressure exerted from outside on a surface of the element of the fluid. In particular, all fluids at rest exhibit this stress behavior, namely that an element of area always experiences a stress normal to itself, and this stress is independent of the orientation. Such stress is called hydrostatic. We call this idealized model a perfect fluid. The equation of motion for perfect fluids is @u C .u r/u D f rp ; @t where .u r/ui D

3 X jD1

uj

@ ui ; @xj

i D 1; 2; 3 :

All real fluids when in motion can exert tangential stresses across surface elements, in which case the tensor T is not diagonal. The stress tensor may always be written in the form Tij D pıij C Pij : In this case Pij is called the viscous stress tensor. In classical fluid dynamics it is assumed that the stress tensor is symmetric, that is, Tij D Tji : This assumption has very important consequences. It may be also considered as a theorem if we assume a specific form of the equation of conservation of angular momentum. We shall discuss this in Sect. 2.2. Exercise 2.8 (Cf. [5]). Show that the Cauchy equation of motion can be written as @ .ui / D fi C .Tji uj ui /;j ; @t and interpret it physically.


19

Exercise 2.9 (Cf. [5]). Show that if F is any function of position and time, then Z Z Dui fi Tji F ;j CF dx FTji nj dS D Dt @˝ ˝ (theorem of stress means). Equation of Energy The first law of thermodynamics in classical hydrodynamics states that the increase of total energy (we shall consider here only kinetic and internal energies) in a material volume is the sum of the heat transferred and the work done on the volume. We denote by q the heat flux (then q n is the heat flux into the volume) and by E the specific internal energy. Then the balance expressed by the first law of thermodynamics is, cf. [5], d dt

Z

1 2 juj C E 2 ˝.t/

dx

(2.18)

Z

Z D ˝.t/

f u dx C

@˝.t/

Z tn u dS

@˝.t/

q n dS :

The first integral on the right-hand side is the rate at which the body force does work, the second integral represents the work done by the stress, and the third integral is the total heat flux into the volume. We shall write this equation in another form. From the theorem of stress means (Exercise 2.9) we have, with F D ui , Z Z Dui fi ui dx : Tji ui;j C ui ui Tji nj dS D Dt @˝.t/ ˝.t/ Rearranging the terms and using the transport theorem, we obtain d dt

Z

1 juj2 dx D 2 ˝.t/

Z ˝.t/

1D 2 juj dx 2 Dt

(2.19)

Z

Z D ˝.t/

fi ui dx

˝.t/

Z Tji ui;j dx C

@˝.t/

ui .tn /i dS :

Thus the rate of change of kinetic energy of a material volume is the sum of three parts: the rate at which the body forces do work, the rate at which the internal stresses do work, and the rate at which the surface stresses do work. From (2.18), (2.19), the transport theorem, and the Green theorem we obtain Z ˝.t/

DE C r q T W .ru/ dx D 0 ; Dt

where T W .ru/ is the dyadic notation for Tji ui;j , the scalar product of T and ru.

20


Thus

DE D r q C T W .ru/ : Dt

Conservation Laws of Classical Hydrodynamics Above we obtained the following system of conservation laws of classical hydrodynamics D D r u ; Dt Du D r T C f ; Dt DE D r q C T W .ru/ : Dt

(2.20) (2.21) (2.22)

They are laws of conservation of mass, momentum, and energy, respectively. If we assume the Fourier law for the conduction of heat, q D kr

.k 0/ ;

(2.23)

where k is the thermal conductivity of the fluid then the energy equation takes the form

DE D r .kr / C T W .ru/ : Dt

2.2 The Stress Tensor In the classical hydrodynamics the stress tensor T is defined by Tij D .p C uk;k /ıij C .ui;j C uj;i / :

(2.24)

If we define the deformation tensor Di;j D

1 .ui;j C uj;i /; 2

(2.25)

then the above formula takes the form Tij D .p C uk;k /ıij C 2 Dij :

(2.26)

Remark 2.1. Formula (2.26) is a consequence of a number of postulates, coming originally from G. Stokes, about the fundamental properties of fluids. These postulates can be formulated as follows (cf. [5, 212]): (a) The stress tensor T is a continuous function of the deformation tensor D and the local thermodynamic state, but independent of other kinematic quantities.

2.2 The Stress Tensor

21

(b) The fluid is homogeneous; that is, T does not depend explicitly on x. (c) The fluid is isotropic; that is, there is no preferred direction. (d) When there is no deformation (D D 0), and the fluid is incompressible (uk;k D 0), the stress is hydrostatic (T D pI, I is the unit matrix). Fluids that satisfy these postulates are called Stokesian. It can be proved (cf. [5, 212]) that the most general form of the stress tensor in this case is T D .p C ˛/I C ˇD C D2 ; where p; ˛; ˇ; are some functions that depend on the thermodynamic state, ˛; ˇ; being dependent as well on the invariants of the tensor D. Moreover, when the dependence of the components of T on the components of D is postulated to be linear, the stress tensor can be written as T D .p C div u/I C 2 D ; which coincides with (2.24). Such linear Stokesian fluids are called Newtonian. Fluids that are not Newtonian are called non-Newtonian. One important example of the latter are the micropolar fluids [92, 159]. The Stress Tensor and the Law of Conservation of Angular Momentum Looking at the form of the equation of conservation of linear momentum d dt

Z

Z ˝.t/

u dx D

Z

˝.t/

f dx C

@˝.t/

tn dS ;

and recalling the definition of angular momentum in mechanics of mass points or rigid particles, it seems natural to assume the following form of the law of conservation of angular momentum: d dt

Z

Z ˝.t/

.x u/ dx D

Z ˝.t/

.x f / dx C

@˝.t/

x tn dS :

(2.27)

In fact, this form of the law of conservation of angular momentum holds if we assume that all torques arise from macroscopic forces. This is the case in most common fluids, but a fluid with a strongly polar character, e.g., a polyatomic fluid, is capable of transmitting stress torques and being subjected to body torques. We call such fluids polar. Theorem 2.1. For an arbitrary continuous medium satisfying the continuity equation (2.9) and the dynamical equation (2.17) the following statements are equivalent: (i) the stress tensor is symmetric, (ii) equation (2.27) holds.

22


Remark 2.2. In classical hydrodynamics the stress tensor is symmetric, and the law of conservation of angular momentum is defined by Eq. (2.27). In consequence, in classical hydrodynamics the law of conservation of angular momentum can be derived from the law of conservation of mass and the law of conservation of linear momentum, and as such adds nothing to the description of the fluid. Proof. Let us assume (ii), and we shall deduce (i). Applying formula (2.11), we have from (2.27) Z d .x u/ dx (2.28) dt ˝.t/ Z D ˝.t/

D .x u/ dx D Dt

Z

Z

D ˝.t/

.x f / dx C

By the Green theorem, Z @˝.t/

Du dx x Dt ˝.t/

Z

@˝.t/

x tn dS :

Z x tn dS D

˝.t/

.x .r T/ C Tx / dx ;

(2.29)

where r T is another notation for div T, and Tx is the vector ijk Tjk ( ijk is the alternating tensor of Levi-Civita), so that by (2.28) Z Z Du f r T dx D x Tx dx : Dt ˝.t/ ˝.t/ The left-hand side vanishes identically by the Cauchy equation; hence the right-hand side vanishes for an arbitrary volume, and so Tx D 0. However, the components of Tx are T23 T32 , T31 T13 , T12 T21 , and the vanishing of these implies Tij D Tji , so that T is symmetric. We leave to the reader the proof that (i) implies (ii). t u

2.3 Field Equations Substituting the stress tensor (2.24) into the system (2.20)–(2.22) we obtain the system of field equations of classical hydrodynamics D D r u ; Dt Du D rp C . C /r div u C u C f ; Dt

(2.30) (2.31)

2.4 Navier–Stokes Equations

23

DE D p div u C ˚ r q ; Dt

(2.32)

where ˚ D .div u/2 C 2 D W D

(2.33)

is the dissipation function of mechanical energy per mass unit. Let us assume that the fluid is viscous and incompressible, namely, that > 0 and div u D 0 ;

(2.34)

that the specific internal energy of the fluid is proportional to its temperature, E D cr ;

where

cr D const > 0 ;

(2.35)

and that Fourier’s law (2.23) (with k D const 0) holds. With (2.34), (2.35), (2.23), and (2.33), system (2.30)–(2.32) becomes @ C u r D 0 ; div u D 0 ; @t @u C .u r/u D rp C u C f ; @t @

cr C u r D 2 D W D C k : @t

(2.36) (2.37) (2.38)

2.4 Navier–Stokes Equations Assuming that the density of the fluid is uniform and denoting D , D is called the kinematic viscosity coefficient), Eqs. (2.36)–(2.38) reduce to @u 1 C .u r/u D rp C u C f ; @t cr

div u D 0 ; @

C u r D 2D W D C : @t

k

(

(2.39) (2.40) (2.41)

24


When the body forces f do not depend on temperature, the first two equations of the above system, @u 1 C .u r/u D rp C u C f ; @t div u D 0

(2.42) (2.43)

constitute a closed system of equations with respect to variables u; p, and are called Navier–Stokes equations of viscous incompressible fluids with uniform density (we shall call them just the Navier–Stokes equations). The mechanical energy of the flow governed by (2.42) and (2.43) due to viscous dissipation is lost and appears as heat. This can be seen from Eq. (2.41) in which the term 2D W D is positive, provided the flow is not uniform. In real fluids, however, density depends on temperature, so that our system (2.39)–(2.41) may be physically impossible. In fact, due to viscosity and high velocity gradients the temperature rises in view of (2.41), and this produces density fluctuations, contrary to our assumption that density is uniform in the flow domain. Thus, reduced problems often play the role of more or less justified approximations. For more considerations of this kind cf. [109, Chap. 1]. When the body forces depend on temperature, f D f . /, we have to take into account the whole system (2.39)–(2.41). One of the considered in the literature system of equations of heat conducting viscous and incompressible fluid are the so-called Boussinesq equations, @u 1 1 C .u r/u D rp C u C g˛. 0 / ; @t 0 0 div u D 0 ;

@

C u r D ; @t cr

(2.44) (2.45) (2.46)

where g represents the vertical gravity acceleration, ˛ is the thermal expansion coefficient, and c r is the thermal diffusion coefficient. Moreover, 0 and 0 are some reference density and temperature, respectively. In the velocity equation the vertical buoyancy force 10 g˛. 0 / results from changes of density associated with temperature variations 0 D ˛. 0 /. This is the only term in the system where changes of density were taken into account. We have also abandoned the viscous dissipation term in the temperature equation.

2.5 Vorticity Dynamics Taking the curl of the equation of motion @u 1 C .u r/u D rp C u; @t

2.5 Vorticity Dynamics

25

we obtain @! C .u r/! D .! r/u C !; @t

(2.47)

where the vector field ! D r u is called vorticity of the fluid. It has a simple physical interpretation. In the case of two-dimensional motion with u D .u1 .x; y/; u2 .x; y/; 0/; the vorticity reduces to @u2 .x1 ; x2 / @u1 .x1 ; x2 / ; ! D .0; 0; !3 .x1 ; x2 // D 0; 0; @x1 @x2 where the third component represents twice the angular velocity of a small (infinitesimal) fluid element at point .x1 ; x2 /. The vorticity field is, by definition, divergence free, div ! D 0: In the case of two-dimensional motions the Eq. (2.47) reduces to @! C .u r/! D !; @t and we can see that the vorticity in the fluid is transported by two agents: convection and diffusion, just as the temperature in the system (2.44)–(2.46). For inviscid fluids ( D 0) the vorticity field has very important properties that allow us to imagine behavior of complicated turbulent flows [83]. In this case, vorticity is a local variable which means that we can isolate a patch of vorticity and observe how it is transported along the velocity field trajectories with a finite speed. For two-dimensional flows this is evident as then the vorticity equation reduces to @! C .u r/! D 0: @t For more information, cf. [166]. Exercise 2.10. Vorticity has nothing in common with rotation of the fluid as a whole. Calculate the vorticity of the flows: (a) u.x1 ; x2 ; x3 / D .u1 .x2 /; 0; 0/ and (b) u.r; ; z/ D .0; k=r; 0/ for r > 0.

26


2.6 Thermodynamics Equations of State From the point of view of thermodynamics the state of a homogeneous fluid can be described by some definite relations among a number of certain state variables, the most important being the volume V .V D 1=/, the entropy S, the internal energy E, the pressure p, and the absolute temperature , cf. [212]. In such a description one may start with a relation of the form (cf. [212]) E D E.S; V/

(Gibbs relation)

(2.48)

and define p and by pD

@E ; @V

D

@E ; @S

(2.49)

with p; > 0 by assumption. In this case, taking the total differential in (2.48) and using (2.49), we obtain dE D dS p dV

or

dE D dS p

1 d : 2

(2.50)

A simple phase system is said to undergo a differentiable process if its state variables are differentiable functions of time: V D V.t/, S D S.t/, etc. Assuming such a dependence one usually assumes, together with (2.50), that DS DV DE D

p Dt Dt Dt or DE DS 1 D D

p 2 : Dt Dt Dt

(2.51)

Relation (2.51) makes it possible to write a definite form of the balance of entropy when we know the laws of conservation of mass and internal energy. We shall use this relation in the sequel. Second Law of Thermodynamics and Constraints on Viscosity Coefficients Consider the law of conservation of energy (2.32)

DE D r .kr / p div u C ˚ ; Dt

(2.52)

where Fourier’s law is assumed, and ˚ is given by (2.33). We see that the internal energy increases with the influx of heat transfer, compression, and the viscous dissipation.

2.6 Thermodynamics

27

From the law of conservation of mass D C div u D 0; Dt we have div u D

1 D : Dt

(2.53)

Substituting (2.53) into (2.52) we can write

DE p D D r .kr / C ˚ : Dt Dt

(2.54)

Now we shall transform the left-hand side of (2.54). From (2.51) we have DS DE p D D C : Dt Dt Dt

(2.55)

DS 1 1 D r .kr / C ˚ ; Dt

(2.56)

Thus so that Z ˝.t/

d DS dx D Dt dt

Z ˝.t/

Z D ˝.t/

S dx

(2.57)

1 1 r .kr / C ˚

dx

k k 1 2 r C 2 jr j C ˚ dx : r D

˝.t/ Z

In the end we obtain the following integral form of the balance of entropy Z ˝.t/

DS dx D Dt

Z @˝.t/

k @

dS C

@n

Z ˝.t/

k 1 2 ˚ dx: jr j C

2

(2.58)

Equation (2.58) with ˚ as in (2.33) and with 0, 3 C 2 0 (cf. [212]) is consistent with the second law of thermodynamics, which says that the rate of increase of entropy is not less than the heat transfer into the material volume. The first integral on the right-hand side of (2.58) is just the heat transferred into the material volume divided by the temperature

28


Z @˝.t/

k @

dS D

@n

Z @˝.t/

q n dS ;

and the second integral on the right-hand side of (2.58) is nonnegative. Remark 2.3. For a perfect fluid, with no heat conductivity and viscosity, Eq. (2.56) becomes DS D 0; Dt so that the entropy is conserved. Remark 2.4. If we assume that the Fourier law holds, the second law of thermodynamics demands k 1 jr j2 C ˚ 0 ; 2

(2.59)

where ˚ is given by (2.33). The inequality (2.59) gives us the restrictions on the coefficients in the constitutive equation (2.24). Without assuming the Fourier law, the second law of thermodynamics states that 1 q r C ˚ 0 :

This law will be satisfied if q r 0 (i.e., the heat flux is not directed against the temperature gradient) and ˚ 0 (i.e., deformation always absorbs energy converting it to heat). Exercise 2.11. Prove that inequality (2.59), with ˚ given in (2.33), yields the following constraints on the coefficients k 0;

3 C 2 0;

0:

Hint. The left-hand side of inequality (2.59) must be nonnegative for an arbitrary choice of ;i and Dij . It is clear that this implies k 0. The nonnegativity of the terms containing Dij is equivalent to the condition 3 C 2 0 and

0:

2.7 Similarity of Flows and Nondimensional Variables Let us consider two initial boundary value problems

Du D rp C u; Dt

div u D 0

(2.60)

2.7 Similarity of Flows and Nondimensional Variables

29

in ˝ D Œ0; Ln , u D u.x; t/, p D p.x; t/, with initial condition u.x; 0/ D u0 .x/, and 0

Du0 D rp0 C 0 u0 ; Dt0

div u0 D 0

(2.61)

in ˝ 0 D Œ0; L0 n , u0 D u0 .x0 ; t0 /, p0 D p0 .x0 ; t0 /, with initial condition u0 .x0 ; 0/ D u00 .x0 /, respectively, and with periodic boundary conditions, n D 2 or 3. Changing dependent and independent variables in (2.60) as follows: u? .x? ; t? / D

u.x; t/ ; U

p? .x? ; t? / D

p.x; t/ p0 U 2

(2.62)

and x? D

x ; L

t? D

U t; L

(2.63)

we obtain the system Du? 1 D rp? C u? ; Dt? Re

div u? D 0

(2.64)

in ˝ ? D Œ0; 1n , u? D u? .x? ; t? /, p? D p? .x? ; t? /, with initial condition u? .x? ; 0/ D u0 .x? / D u0 .Lx? /=U, with periodic boundary condition, and with Re D

LU :

In the case of problem (2.60) the symbols ; L; U; denote dimensional characteristic quantities of the flow, describing the density of the fluid (), characteristic linear dimension of the domain of the flow (L), some characteristic velocity (U), and viscosity ( ). For U we can take, for example, the square root of the average initial kinetic energy in ˝, see [88] for more details. According to (2.62) and (2.63), in (2.64) all dependent and independent variables are nondimensional, together with the Reynolds number Re. The characteristic quantities of the flow in (2.64) are equal to one, with the exception of viscosity which equals 1=Re. In the same way we can reduce system (2.61) to system (2.64), now with the Reynolds number Re0 D

0 L0 U 0 : 0

Assume now that Re D Re0 . Then both systems (2.60) and (2.61) (including the boundary conditions) are represented, in nondimensional variables, by the same system (2.64). We call systems (2.60) and (2.61) dynamically similar. Solving system (2.64) we generate solutions to an infinite three-parameter family of systems having the same Reynolds number.

30


The physical interpretation of the Reynolds number is that it describes the relation between the magnitudes of the inertial term .u r/u and the viscous term u ( D =) in the equation of motion, according to inertial term j.u r/uj U.U=L/ DO DO D O.Re/; viscous term juj .U=L2 / or, saying it differently, the Reynolds number can be interpreted as a ratio of the strength of inertial forces to viscous forces. Exercise 2.12. Let u? ; p? solve system (2.64). Show that u.x; t/ D Uu?

x U ; t ; L L

p.x; t/ D U 2 p?

x U ; t C p0 L L

solve system (2.60). Exercise 2.13. Let D 0 and D 0 , and let u0 ; p0 solve system (2.61). Show that U 0 L0 L0 U x; u.x; t/ D 0 u t ; U L LU 0 0 L0 U U2 0 L 0 x; t p0 C p0 p.x; t/ D 02 p U L LU 0 solve system (2.60). Here arbitrary constants p0 ; p00 may represent typical values of pressures in the considered problems for systems (2.60) and (2.61), respectively, cf. (2.62). Exercise 2.14. Let u0 ; p0 solve the system of equations 0

Du0 D rp0 C u0 ; Dt

div u0 D 0

(2.65)

in the whole space. Show that u.x; t/ D lu0 .lx; l2 t/;

p.x; t/ D l2 .p0 .lx; l2 t/ p00 / C p0 ;

where l > 0, also solve this system of equations. Thus having one (nontrivial) solution of system (2.65) we can generate in principle an infinite number of different solutions of the same system. Consider the transformations, 1 1 (2.66) .x; t; u; p/ ! lx; l2 t; u; 2 p ; l > 0; l l

2.8 Examples of Simple Exact Solutions

31

or x0 D lx ; t 0 D l2 t ; 1 u; l 1 p0 D 2 p: l

u0 D

Show that transformations (2.66) form a group. This group is called a one parameter symmetry group of scalings of system (2.65). A solution .u0 ; p0 / is called self-similar if u.x; t/ D u0 .x0 ; t0 /; p.x; t/ D p0 .x0 ; t0 /, cf. [17]. Exercise 2.15. Let u0 ; p0 solve the system of equations 0

Du0 D rp0 C u0 ; Dt

div u0 D 0

in the whole space Rn . Show that u.x; t/ D u0 .x ct; t/ C c; p.x; t/ D p0 .x ct; t/; where c is any vector in Rn , also solve this system of equations (Galilean invariance), and u.x; t/ D Qt u0 .Qx; t/; p.x; t/ D p0 .Qx; t/; where Q is any rotation matrix (Qt D Q1 ) also solve this system of equations (rotation symmetry).

2.8 Examples of Simple Exact Solutions The Flow Due to an Impulsively Moved Plain Boundary Let the domain of the flow be half-space f.x; y; z/ 2 R3 W y > 0g; and assume that at times t < 0 the flow is at rest, and that at time t D 0 the plane y D 0 is suddenly jerked into motion in the x-direction, with a constant velocity U.

32


Due to the geometrical symmetry of the problem and under additional “intuitively apparent” preconceptions, we assume that the flow is of the form u D .u.y; t/; 0; 0/. Substituting u of the above form into the Navier–Stokes equations (2.42) we obtain @u 1 @p @2 u D C 2 ; @t @x @y

(2.67)

@p @p D D 0: @y @z

(2.68)

As u depends only on y, from (2.67) together with (2.68) it follows that @p depends @x only on t. Assume additionally that @p D 0 and that the flow is at rest for y D 1. @x Then the above system together with the initial and boundary conditions reads @u @2 u D 2; @t @y

(2.69)

with u.y; 0/ D 0

for

y > 0;

(2.70)

u.0; t/ D U

for

t > 0;

(2.71)

u.1; t/ D 0

for

t > 0:

(2.72)

We say that a solution u D u.y; t/ of the initial and boundary value problem (2.69)– (2.72) is invariant or self-similar with respect to the one-parameter symmetry group of scalings y0 D ˛y;

t0 D ˛ 2 t;

u0 D u;

˛ > 0;

(2.73)

if the function u0 D u0 .y0 ; t0 / D u.y; t/ is a solution of the problem @u0 @2 u0 D ; @t0 @y02 with u0 .y0 ; 0/ D 0 for u0 .0; t0 / D U

y0 > 0 ;

for t0 > 0 ;

u.1; t0 / D 0 for

t0 > 0:

Observe that the transformation of the independent variables does not change the space-time domain.


33

The relation u0 .y0 ; t0 / D u.y; t/ satisfied by the invariant solution implies that the number of its independent variables can be reduced by one [17]. Together with the dimensional homogeneity principle [33] stating that physical phenomena must be described by laws that do not depend on the unit of measure applied to the dimensions of the variables that describe the phenomena, we arrive at the following form of the solution y D p : t

u.y; t/ D f ./;

(2.74)

Notice that the new variable , called similarity variable is nondimensional and is invariant with respect to the symmetry group of scalings y0 D ˛y, t0 D ˛ 2 t. Exercise 2.16. Assume that problem (2.69)–(2.72) is invariant with respect to a dilation group y0 D ea y;

t0 D eb t;

u0 D ec u;

for some a; b; c, that is if u.x; t/ is a solution of the problem then u0 .x0 ; t0 / is also a solution of the same problem. Then we get (2.73). Observe, that if a problem is uniquely solvable and invariant with respect to a symmetry group of scalings x ! x0 ; t ! t0 ; u ! u0 then its unique solution is also invariant. Substituting function of the form (2.74) to problem (2.69)–(2.72) we obtain 1 f 00 C f 0 D 0; 2 with f .0/ D U;

f .1/ D 0:

Integrating this equations we obtain Z 1 s2 u.y; t/ D f ./ D U 1 p e 4 ds ; 0 or, in the nondimensional form u.y; t/ 1 D1 p U

Z

e

s2 4

ds:

0

The example of the velocity profile given by (2.75) is depicted in Fig. 2.1.

(2.75)

34


h=

y 3,5 vt 3 2,5 2 1,5 1 0,5 0 0

u(h) 0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

U

Fig. 2.1 Velocity profile for the flow in the half-space due to the impulsively moved boundary. In the plot we have U D 1. The variable on the horizontal axis is the velocity u, and on the vertical y axis D pt

Exercise 2.17. Check that the pair .u; p/ D .u..y; t/; 0; 0/; C/ where C is a constant and u is the function given in (2.75) is a solution of the Navier–Stokes equations (2.42) and (2.43) with initial and boundary conditions (2.70)–(2.72). Observe that we have found only one of, perhaps, many other solutions of the problem. We cannot claim that the solution is unique in u. From (2.75) it follows that the velocity of the flow is a nonlocal variable, that is, it spreads information through the region of the flow with infinite speed. Although at negative times the flow was at rest, at any, however small, positive time t and any, however large, y > 0, we have u.y; t/ > 0. The same observation concerns the vorticity vector (the third and only nonzero component of which is ! D @u ) as @y both functions u and ! satisfy the diffusion equation. The Poiseuille Flow Consider a stationary flow u D .u.y/; 0; 0/ in the domain f.x; y; z/ 2 R3 W 0 < y < hg; and with homogeneous boundary conditions u D 0 at y D 0 and y D h. Substituting u of the above form to the Navier–Stokes equations we obtain the equations of motion

1 @p @2 u ; D 2 @y @x @p @p D D 0; @y @z

from which it follows that

@2 u 1 @p DC D 2 @y @x

(2.76)


35

0,014

0,012

0,01

u(y)

0,008

0,006

0,004

0,002

0 0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

y

1

h=1

Fig. 2.2 Parabolic velocity profile for the Poiseuille flow. The constants were chosen as h D 1, D 10, and C D 1. Velocity u.y/ is on the vertical axis, and the space variable y is on the horizontal axis

for some constant C. Taking into account the boundary conditions we obtain u.y/ D

C y.y h/ ; 2

where CD

1 @p : @x

From the last equation we have p.x/ D Cx C B, for an arbitrary constant B. Let us take B D 0. We can see that there is an infinite number of solutions .u; p/ of the form u.y/ D

C y.y h/; 2

p.x/ D Cx:

(2.77)

We call them the Poiseuille flows. For C D 0 the flow is just at rest, both velocity and pressure equal zero. For C < 0 the fluid flows towards the increasing x-direction, u > 0, driven by the force due to the pressure gradient (as C < 0 the pressure decreases with increasing x). The example of the velocity profile given by (2.77) for negative value of C is depicted in Fig. 2.2.

36


U 2 1,8 1,6 1,4 1,2

u(y)

1 0,8 0,6 0,4 0,2 0

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

y1

h=1

Fig. 2.3 Linear velocity profile for the Couette flow. The constants were chosen as h D 1 and C D 2. Velocity u.y/ is on the vertical axis, and the space variable y is on the horizontal axis

The Couette Flow Let us consider a flow u D .u.y/; 0; 0/ in the same region as in the preceding example, this time, however, with different boundary conditions. We pose u D 0 at y D 0 and u D U at y D h. As the driving force now comes from the motion of the upper boundary y D h, we pose C D 0 which corresponds to p D const in the whole domain. Solving Eq. (2.76) in u with C D 0 and with the new boundary conditions we obtain the solution u.y/ D

U y; h

p D const:

(2.78)

of the problem, with a uniform pressure distribution. We call this flow the Couette flow. The velocity profile given by (2.78) for negative value of C is depicted in Fig. 2.3.

2.9 Comments and Bibliographical Notes Remarks on Modeling Exact solutions serve, among other things, to test a given theory of hydrodynamics. By theory we mean here a set of governing equations (conservation laws) together with additional conditions (boundary or initial and boundary conditions), that is, a boundary or initial and boundary value problem. In the following chapters we shall prove several results about such theories of classical hydrodynamics. A particular theory of hydrodynamics is well-set, that is, that it possesses a unique solution depending continuously upon the data (boundary data, external forces, etc.).

2.9 Comments and Bibliographical Notes

37

As was observed by Birkhoff in [16], “until it has been shown that the boundary value problem is well-set, one cannot conclude that its equations are erroneous.” The latter means “false” according to the following definition, cf. [16]: A theory of rational hydrodynamics will be called incomplete if its conditions do not uniquely determine the flow, overdetermined if its conditions are mathematically incompatible, false if it is well-set but gives grossly incorrect predictions. Since from a particular theory in its whole generality, it is in most cases quite impossible to predict qualitative properties of the flow (it is sometimes easier, however, to compare various flows among themselves), we are forced to “simplify,” whatever that means. In this connection, there are very often exact solutions of special problems that serve to test whether a given theory gives correct predictions and consequently is not a false theory. We enumerate a number of such simplifications (which we have already used in Sect. 2.8): 1. linearization of governing equations, 2. setting a special form of boundary conditions, 3. assumption of some symmetry in space (which reduces the number of spatial dimensions), 4. assumption that the problem is time-independent, 5. assumption that acting forces are potential or absent. After suitable simplifications have been done, the original problem reduces to the one that can be (luckily) solved explicitly by solving, e.g., a linear system of ordinary differential equations with constant coefficients. Having the exact solutions, we try to compare them with experiments. When simplifying, however, one has to be very careful in order not to oversimplify and obtain paradoxes, for example. It is well known (cf. [16]) that the various plausible intuitive assumptions we make implicitly when simplifying may be fallacious. Following [16], we state the assumptions that are especially suggestive: 1. intuition suffices for determining which physical variables require consideration, 2. small causes produce small effects, and infinitesimal causes produce infinitesimal effects, 3. symmetric causes produce effects with the same symmetry, 4. the flow topology can be guessed by intuition, 5. the processes of analysis can be used freely: the functions of rational hydrodynamics can be freely integrated, differentiated, expanded in series, integrals, etc., 6. mathematical problems suggested by intuitive physical ideas are well-set. In Chaps. 14–16 we consider problems whose solutions might not be unique. For more information about classical hydrodynamics we refer the reader to [1, 14, 83, 149]. For a history of hydrodynamics and the derivation and development of the Navier–Stokes equations, in particular, see [81].

3

Mathematical Preliminaries

In this chapter we introduce the basic preliminary mathematical tools to study the Navier–Stokes equations, including results from linear and nonlinear functional analysis as well as the theory of function spaces. We present, in particular, some of the most frequently used in the sequel embedding theorems and differential inequalities.

3.1 Theorems from Functional Analysis Theorems from Linear Functional Analysis The core of this subsection is the Lax–Milgram lemma (which we state for separable Hilbert spaces) as well as its proof using the Galerkin method. This method will be very useful in our subsequent considerations, in the proofs of the existence of solutions of both linear and nonlinear problems. Theorem 3.1. In a reflexive Banach space every closed ball is weakly compact. Proof. A proof can be found, e.g., in [106], for separable spaces in [118].

t u

In particular, let fxn g be a sequence in a reflexive Banach space B such that kxn kB M for some M > 0. Then there exists a subsequence fx g of the sequence fxn g and an element x in B with kxkB M such that fx g converges weakly to x in B; that is, for every linear and continuous functional L on B, lim L.x / D L.x/ :

!1


39

40

3 Mathematical Preliminaries

Theorem 3.2 (Lax–Milgram Lemma for Separable Spaces). Let H be a separable Hilbert space with a norm kk, L 2 H 0 a linear functional on H, a.u; v/ a bilinear and continuous form on H H, coercive, that is, such that for some ˛ > 0 and for all u 2 H a.u; u/ ˛kuk2 : Then there exists a unique element u 2 H such that a.u; v/ D L.v/

for all

v 2 H:

(3.1)

Proof. We use the Galerkin method, to which we shall refer in the following chapters. It consists of proving the existence of elements um 2 Hm such that a.um ; v/ D L.v/

for all

v 2 Hm ;

(3.2)

where HS m are finite dimensional subspaces of H such that H1 H2 Hm and 1 mD1 Hn is a dense subset of H, and then by passing to the limit with n to obtain (3.1). Let !1 , !2 , !3 , : : : be a basis of H and Hm D spanf!1 ; : : : ; !m g, m D 1; 2; 3; : : : . 1. We shall show that for each positive integer m there exists um 2 Hm such that (3.2) holds. Let um D

m X

k !k :

kD1

Then (3.2) is equivalent to the system of linear equations m X

k a.!k ; !l / D L.!l / ;

l D 1; 2; : : : ; m :

kD1

This system has a unique solution .1 ; : : : ; m / for every right-hand side if and only if the matrix fa.!k ; !l /gk;lm is nonsingular. We shall show that the homogeneous system m X

k a.!k ; !l / D 0 ;

l D 1; 2; : : : ; m ;

kD1

has a unique solution D .0; 0; : : : ; 0/. We multiply the lth equation by k and add the equations to get m X m X lD1 kD1

l k a.!k ; !l / D a

m X kD1

k !k ;

m X lD1

! l !l

D a.um ; um / D 0 :

3.1 Theorems from Functional Analysis

41

From the coerciveness of the form a.; / we obtain um D 0. As the vectors !1 ; : : : ; !m are linearly independent, we conclude that 1 D 2 D D m D 0. Hence the matrix fa.!k ; !l /gk;lm is nonsingular, and for each positive integer m there exists an approximate solution um 2 Hm . 2. Convergence of the sequence fum g. We have ˛kum k2 a.um ; um / D L.um / kLkH 0 kum k ; whence kum k

1 kLkH 0 ˛

for each positive integer m. From Theorem 3.1 it follows that there exists a subsequence fu g of the sequence of approximate solutions fum g and an element u 2 H such that u !u weakly in H. For j we have a.u ; v/ D L.v/

for all

v 2 Hj H ;

so that lim a.u ; v/ D a.u; v/ D L.v/

!1

for

v2

1 [

Hj :

jD1

S As 1 jD1 Hj is a dense subset of H, we conclude from the continuity of a.; / and L./ that (3.1) holds for every v 2 H. 3. Uniqueness of u. Let us suppose that we have two different elements u1 and u2 such that a.u1 ; v/ D L.v/ and a.u2 ; v/ D L.v/ for all v 2 H. Thus a.u1 u2 ; v/ D 0, and taking v D u1 u2 we obtain 0 D a.u1 u2 ; u1 u2 / ˛ku1 u2 k2 ; whence u1 D u2 . We have come to a contradiction, which proves the uniqueness. t u Remark 3.1. If L is a linear and continuous functional on the Banach space B, we will write L 2 B0 , the dual space of B, and we will use the notation L.v/ D hL; vi for v 2 B. Exercise 3.1. Prove that the whole sequence of approximate solutions converges weakly to u. As a corollary we obtain the following Theorem 3.3 (Riesz–Fréchet Theorem for Separable Spaces). Let H be a separable Hilbert space with scalar product .; / and norm k k and let L 2 H 0 be a linear and continuous functional on H. Then there exists a unique element u 2 H such that .u; v/ D L.v/ and kuk D kLkH 0 .

for each

v2H

42


Remark 3.2. In the above two theorems the separability of the space H is not essential; for the proofs see [110]. Fixed Point Theorems Fixed point theorems are the basic tool that we shall use. We begin with the Banach contraction principle, the only theorem in this subsection that guarantees the uniqueness of the fixed point. The existence of a unique fixed point is a strong condition, however, and in most cases we shall make use of the Schauder or the Leray–Schauder theorems. They both follow from the Brouwer fixed point theorem. Theorem 3.4 (Banach Contraction Principle). Let T be an operator defined on the Banach space X with values in X. Assume that T is a contraction, i.e., that a number ˛, 0 ˛ < 1, exists such that for all pairs of elements u; v 2 X we have kTu TvkX ˛ku vkX : Then there exists a unique element u 2 X such that Tu D u. Proof. Let u0 be an arbitrary element of the space X. Then the sequence u0 ; u1 ; u2 ; : : :, where for n 1, un are defined recursively by un D Tun1 , converges in X to the fixed point of the operator T. t u Exercise 3.2. Provide the details of the proof and show that kun ukX

˛ n1 ku1 u0 kX ; 1˛

for n D 1; 2; : : : . Theorem 3.5 (Brouwer). Let K be a nonempty, convex, and compact set in Rn . If T W K!K is a continuous mapping, then it has at least one fixed point, that is, there exists u0 2 K such that T.u0 / D u0 . t u

Proof. For a proof see, e.g., [118, 158].

Theorem 3.6 (Schauder). Let K be a closed, nonempty, bounded, and convex set in a Banach space X. Let T be a completely continuous (that is, compact and continuous) operator defined on K such that T.K/ K : Then there exists at least one element u0 2 K such that T.u0 / D u0 . Remark 3.3. If X D Rn , then the Schauder theorem reduces to the Brouwer theorem. Proof. The closure of the set T.K/ is compact. For each n 2 N there exists a sequence x1 , x2 , : : :, xr.n/ in T.K/ such that min kx xi kX < " D

1ir.n/

1 n

for each

x 2 T.K/ :

3.1 Theorems from Functional Analysis

43

Let Xn be the linear hull of the set fx1 ; : : : ; xr.n/ g, that is, the set of all points of the form 1 x1 C : : : C r.n/ xr.n/ , where 1 , : : :, r.n/ are arbitrary real numbers. We will write Xn D span fx1 ; : : : ; xr.n/ g. Set Kn D K \ Xn . The set Kn is convex, bounded, closed, nonempty, and lies in a finite dimensional subspace of X. We define the map Tn W K!Kn by Tn .x/ D F 1 T.x/ ; n

where Pr.n/

mi .x/xi ; F" .x/ D PiD1 r.n/ iD1 mi .x/

mi .x/ D

8 < " kx xi kX when :

0

kx xi kX < " ; kx xi kX " :

when

We have F" .K/ Kn . The map Tn is continuous, as the functions mi are continuous. For an arbitrary x 2 K we have P P mi .Tx/Tx mi .Tx/xi P kTx Tn xkX D m .Tx/ i

P

X

1 mi .Tx/kTx xi kX P < : mi .Tx/ n

(3.3)

Thus, we have approximated in a uniform way the compact operator T by a finite dimensional operator Tn . Since Tn W Kn !Kn satisfies the hypothesis of the Brouwer fixed point theorem, there exist xN n such that Tn xN n D xN n . As T is compact, the sequence fNxn g has a convergent subsequence xN nk !Nx. From the continuity of T it follows that T xN nk !T xN . From (3.3) we conclude that T xN nk Tnk xN nk !0. Thus T xN D xN , and T has a fixed point in K. t u Exercise 3.3. Prove, by giving simple examples, that all assumptions in the Schauder theorem are necessary. Theorem 3.7 (Leray–Schauder). Let T be a completely continuous mapping of a Banach space X into itself, and suppose that there exists a constant M such that kxkX < M for all x 2 X and 2 Œ0; 1 satisfying x D Tx. Then T has a fixed point.

(3.4)

44


Proof (cf. [110]). We may assume without loss of generality that M D 1. Let us define the map T by 8 Tx if kTxkX 1 ; ˆ ˆ < T x D Tx ˆ ˆ : if kTxkX > 1 : kTxkX Then, T is a continuous mapping of the closed unit ball B.1/ D fx 2 X W kxkX 1g into itself. Since the set TB.1/ is precompact, the same is true for T B.1/. From the Schauder fixed point theorem it follows that T has a fixed point x. We shall show that x is a fixed point of T. In fact, if kTxkX > 1, then x D T x D Tx 1 with D kTxk and kxkX D kT xkX D 1, which contradicts (3.4). t u X In Chaps. 5 and 15 we will use the following Kakutani–Fan–Glicksberg fixed point theorem (see [3], Corollary 17.55) to prove the existence of weak solutions. Theorem 3.8 (Kakutani–Fan–Glicksberg). Let S X be nonempty, compact, and convex set, where X is a locally convex Hausdorff topological vector space and let the multifunction ' W S ! 2S have nonempty convex values and closed graph. Then the set of fixed point of ' (i.e., fx 2 S W x 2 '.x/g) is nonempty and compact.

3.2 Sobolev Spaces and Distributions We assume that ˝ is a nonempty open subset of Rn , n a positive integer. Definition 3.1. By Lp .˝/, 1 p < 1, we denote a linear space of real-valued functions f defined on ˝ such that jf jp is integrable on ˝ with respect to Lebesgue measure. The functional Z f 7! kf kLp .˝/ D

jf jp dx

1=p

˝

is a norm in Lp .˝/ (provided that we identify functions that are equal to each other almost everywhere in ˝). The space Lp .˝/ is a Banach space. For p D 2 it is a Hilbert space with the scalar product Z .f ; g/ D fg dx: ˝

We have Theorem 3.9. Let ffn g be a Cauchy sequence in Lp .˝/, that is, lim kfm fn kLp .˝/ D 0:

m;n!1

3.2 Sobolev Spaces and Distributions

45

Then there exists f 2 Lp .˝/ and a subsequence ff g of the sequence ffn g such that lim kf f kLp .˝/ D 0

!1

(completeness of Lp .˝/), and f .x/ ! f .x/ for almost all x 2 ˝. Moreover there exists h 2 Lp .˝/ with h.x/ 0 a.e. in ˝ such that jf .x/j h.x/ for a.e. x 2 ˝. Definition 3.2. By L1 .˝/ we denote the set of measurable real functions f on ˝ for which ess supfjf .x/j W x 2 ˝g inffk > 0 W .fx 2 ˝ W jf .x/j > kg/ D 0g < 1: The linear space L1 .˝/ with norm kf kL1 .˝/ D ess supfjf .x/j W x 2 ˝g is a Banach space. We say that a real-valued function f on ˝ is locally integrable on ˝, and write 1 f 2 Lloc .˝/, if for every compact K ˝, f is integrable on K. p Similarly, we define spaces Lloc .˝/, for 1 < p < 1. For f defined on ˝ we define the support of f as the closure of the set fx 2 ˝ W f .x/ ¤ 0g, and denote it by supp f . Let ˛ D .˛1 ; : : : ; ˛n / be a multi-index the coordinates of which are nonnegative integers. We write j˛j D ˛1 C C ˛n , and D˛ D

@j˛j : @˛1 x1 @˛n xn

We say that f 2 Ck .˝/ (resp. f 2 Ck .˝/) if there exist partial derivatives D˛ f for all ˛ with j˛j k that are continuous in ˝ (resp. ˝). Moreover, C1 .˝/ D

1 \

Ck .˝/:

kD1

By C01 .˝/ we denote the set of f 2 C1 .˝/ for which supp f ˝. If ˝ is bounded, supp f is a compact subset of ˝. Lemma 3.1 (cf. [2, 142]). The set C01 .˝/ is dense in Lp .˝/ for 1 p < 1. 1 Lemma 3.2 (Du Bois–Reymond, cf. [2, 182]). Let f 2 Lloc .˝/, and let

Z ˝

f ' dx D 0

for each ' 2 C01 .˝/. Then f D 0 almost everywhere in ˝.

46


Definition 3.3 (Generalized Derivative). Let ˛ D .˛1 ; : : : ; ˛n / be a multi-index 1 the coordinates of which are nonnegative integers. We call a function f ˛ 2 Lloc .˝/ 1 the ˛th generalized (or weak) derivative in ˝ of f 2 Lloc .˝/ if for each ' 2 C01 .˝/ Z Z fD˛ ' dx D .1/j˛j f ˛ ' dx: ˝

˝

Exercise 3.4. Using Lemma 3.2 show that if f 2 Cj˛j .˝/, then f ˛ D D˛ f ; that is, there exists the generalized derivative f ˛ , and it equals the usual classical partial derivative D˛ f . Exercise 3.5. Prove that generalized partial derivatives are uniquely determined (provided that they exist). Exercise 3.6. Let ˝ be the interval .0; 1/ on the real line, and let f .x/ D jxj. Prove that f has the generalized first derivative f 0 , and f 0 .x/ D sgn x almost everywhere in ˝. In the sequel we shall write D˛ f instead of f ˛ for generalized derivatives. Definition 3.4 (Sobolev Space W m;p .˝/). Let m be a positive integer, 1 p < 1. By W m;p .˝/ we denote a linear subspace of elements f in Lp .˝/ for which generalized partial derivatives D˛ f exist for all j˛j m and belong to Lp .˝/, with norm 11=p 0 X p kf kW m;p .˝/ D @ kD˛ f kLp .˝/ A : (3.5) j˛jm

Lemma 3.3 (cf. [2, 142]). The space W m;p .˝/ is a Banach space. For 1 < p < 1 W m;p .˝/ is reflexive. Exercise 3.7. Prove that the space W m;p .˝/ is complete. Hint. Use Theorem 3.9, Lemma 3.2, and Definition 3.4. Lemma 3.4. For 1 < p < 1 every bounded sequence in W m;p .˝/ contains a weakly convergent subsequence. Proof. The space W m;p .˝/ is reflexive. Definition 3.5. By of W m;p .˝/.

m;p W0 .˝/

t u

we denote the closure of the set

C01 .˝/

in the norm

The spaces W m;p .˝/ are also denoted by Wpm .˝/. The spaces W m;2 .˝/ and denoted also by H m .˝/ and H0m .˝/, respectively, are Hilbert spaces with scalar product W0m;2 .˝/,

.f ; g/ D

X Z j˛jm

D˛ fD˛ g dx: ˝

3.2 Sobolev Spaces and Distributions

47

Definition 3.6. Let m be a positive integer, 1 < p < 1, and 1=p C 1=q D 1. By W m;q .˝/ we denote the space of linear continuous functionals on the space m;p W0 .˝/. The spaces W m;2 .˝/ are also denoted by H m .˝/. Now we shall define distributions and distributional derivatives. Let us introduce in the set C01 .˝/ the following notion of convergence. Definition 3.7. We say that a sequence f'n g C01 .˝/ converges to zero if there exists a compact K ˝ such that (i) supp 'n K for each 'n , (ii) lim D˛ 'n D 0 for each multi-index ˛, uniformly on ˝. n!1

We denote by D.˝/ the set C01 .˝/ together with the convergence introduced above. Definition 3.8. We call the map T W D.˝/ ! R a distribution on ˝ if it is linear, that is, T.˛' C ˇ / D ˛T.'/ C ˇT. / for all ˛; ˇ 2 R and ';

2 D.˝/, and if lim T.'n / D 0

n!1

for each sequence f'n g D.˝/ that converges to zero in D.˝/. We denote by D 0 .˝/ the set of all distributions on ˝ and write hT; 'i instead T.'/. 1 For f 2 Lloc .˝/, the map on D.˝/ defined by Z hTf ; 'i D

˝

f ' dx

is a distribution. By Lemma 3.2 we can identify it with f , and we write Z hf ; 'i D

˝

f ' dx :

Distributions that can be represented by locally integrable functions we call regular. An example of a distribution that is not regular is the map ı.x0 / W D.˝/ 3 ' ! ıx0 .'/ D '.x0 / 2 R; for some x0 2 ˝, called the Dirac delta.

48


Definition 3.9 (Distributional Derivative). Let T 2 D 0 .˝/. Then the map @T W D.˝/ ! R; @xi i 2 f1; : : : ; ng; defined by

@T @' ; ' D T; @xi @xi

for each ' 2 D.˝/, is a distribution. Similarly, for an arbitrary multi-index ˛ we define D˛ T 2 D 0 .˝/ by hD˛ T; 'i D .1/j˛j hT; D˛ 'i for each ' 2 D.˝/. For T 2 D 0 .˝/, we call D˛ T the ˛th distributional derivative of T. Exercise 3.8. Prove that each distribution has all distributional derivatives of any order. Exercise 3.9. Show that f 00 .x/ D 2ı.x/ for f from Exercise 3.6 above. One can show that there is no locally integrable function g on ˝ such that f 00 D g. Definition 3.10. We say that an open and bounded subset ˝ Rn belongs to the class Ck;˛ , 0 ˛ 1, k a nonnegative integer, if for every point x0 2 @˝ there exists a ball B centered at x0 and a one-to-one mapping of B on D Rn such that (i) (ii) (iii)

.B \ ˝/ RnC D fx D .x1 ; : : : ; xn / 2 Rn W xn > 0g , .B \ @˝/ @RnC , 2 Ck;˛ .B/, 1 2 Ck;˛ .D/.

If the boundary of ˝ is smooth enough, say ˝ 2 C0;1 , then the Sobolev spaces W .˝/ defined as in Definition 3.4 above can be described alternatively as the closure of the set Cm .˝/ in the norm (3.5), cf. [142]. From this new definition of W m;p .˝/ it is clear that the set Cm .˝/ is dense in W m;p .˝/ (provided that the boundary of ˝ is smooth enough). Still another description, equivalent to Definition 3.4, of Sobolev spaces can be given in terms of distributions. We define the space W m;p .˝/ as the linear subspace of those f 2 Lp .˝/ for which all distributional derivatives D˛ f , j˛j m, belong to Lp .˝/, with norm (3.5). m;p

3.3 Some Embedding Theorems and Inequalities We begin with three general results. Theorem 3.10 ([110, Theorem 7.26], [185]). Let ˝ be an open and bounded subset of Rn with Lipschitz boundary, that is, ˝ 2 C0;1 . Then

3.3 Some Embedding Theorems and Inequalities

49

if kp < n, then the space W k;p .˝/ is continuously embedded in the space L .˝/, p D np=.n kp/, and compactly embedded in Lq .˝/ for q < p ; (ii) if 0 m < kn=p < mC1, then the space W k;p .˝/ is continuously embedded in Cm;˛ .˝/, ˛ D k n=p m, and compactly embedded in Cm;ˇ .˝/, for ˇ < ˛. (i)

p

Lemma 3.5 (A Version of the Rellich Theorem, cf. [182]). Let ˝ be an open and bounded subset of Rn . Then the embedding of H01 .˝/ in L2 .˝/ is compact. Theorem 3.11 ([105, Theorem 10.1], [235, p. 1034]). Let ˝ be a bounded domain in Rn with Lipschitz boundary, and let u be any function in W m;r .˝/ \ Lq .˝/, 1 r; q 1. For any multi-index j, 0 jjj < m, and for any number from the interval jjj=m 1, set 1 1 m jjj 1 D C

C .1 / : p m r n q If m jjj n=r is not a nonnegative integer, then kDj ukLp .˝/ Ckuk W m;r .˝/ kuk1

Lq .˝/ :

(3.6)

If m jjj n=r is a nonnegative integer, then inequality (3.6) holds for D jjj=m. The constant C depends only on ˝, r, q, m, j, . Now we shall state some special cases of Theorems 3.10 and 3.11, useful in the following chapters. Let ˝ R3 be as in Theorem 3.10. Then 1. H 1 .˝/ is continuously embedded in L6 .˝/, kukL6 .˝/ CkukH 1 .˝/ (cf. Lemma 3.9 below). 2. H 1 .˝/ is compactly embedded in L4 .˝/. 3. H 2 .˝/ is continuously embedded in C0;1=2 .˝/, and C0;1=2 .˝/ is compactly embedded in C.˝/; in particular, we have ess sup ju.x/j CkukH 2 .˝/ : x2˝

Exercise 3.10. Show that W 2;3=2 .˝/ is continuously embedded in W 1;3 .˝/ for ˝ R3 as in Theorem 3.10. Let ˝ be a bounded domain in R3 with Lipschitz boundary. Then 3=4

1=4

kukL4 .˝/ CkukH 1 .˝/ kukL2 .˝/ [cf. inequality (3.10)].

50


Lemma 3.6 (Poincaré Inequality). Let ˝ be an open and bounded subset of Rn , n a positive integer, d D diam ˝ D supfjxyj W x; y 2 ˝g. Then, for each u 2 H01 .˝/ and i 2 f1; : : : ; ng, d @u : (3.7) kukL2 .˝/ p 2 @xi L2 .˝/ Proof. Let u 2 C01 .˝/, and fix i D 1. We have Z x1 @u u.x1 ; : : : ; xn / D .t; x2 ; : : : ; xn /dt: 1 @t By the Schwarz inequality, Z ju.x1 ; : : : ; xn /j

x1

x1

Z

ˇ ˇ2 ˇ @u ˇ ˇ .t; x2 ; : : : ; xn /ˇ dt .x1 x / 1 ˇ @t ˇ

C1

1

ˇ ˇ ˇ @u ˇ2 ˇ ˇ ˇ @x .x/ˇ dx1 .x1 x1 / ; 1

x1

where D infft W u.t; x2 ; : : : ; xn / D 0; .t; x2 ; : : : ; xn / 2 ˝g: Integrating with respect to x1 we obtain ˇ ˇ Z C1 Z d2 C1 ˇˇ @u ˇˇ2 2 ju.x/j dx1 .x/ dx1 : 2 1 ˇ @x1 ˇ 1 Now, integrating with respect to x2 ; : : : ; xn we obtain inequality (3.7) for i D 1 and u 2 C01 .˝/. Let u 2 H01 .˝/. There exists a sequence fun g C01 .˝/ converging to u in H01 .˝/. We prove (3.7) for u, passing to the limit in @un d ; kun kL2 .˝/ p 2 @xi L2 .˝/ for i 2 f1; : : : ; ng.

t u

Corollary 3.1. Let ˝ R be an open and bounded set. Then the following two norms are equivalent in H01 .˝/: 11=2 0 X kD˛ uk2L2 .˝/ A ; kuk D @kuk2L2 .˝/ C n

j˛jD1

and 0 jjjujjj D @

X

j˛jD1

11=2 kD˛ uk2L2 .˝/ A

:


51

Lemma 3.7 (Ladyzhenskaya Inequality in Two Dimensions). Let ˝ R2 be an open and bounded set. Then for all u 2 H01 .˝/ 1=2

1=2

kukL4 .˝/ 21=4 kukL2 .˝/ kDukL2 .˝/ :

(3.8)

Proof (cf. [146]). We shall prove inequality (3.8) for smooth functions with compact support in ˝, and the general case will follow immediately from the density of these functions in H01 .˝/. Let u 2 C01 .˝/. For convenience we consider u as defined on the whole space R2 and equal zero outside of ˝. We have Z xk u2 .x1 ; x2 / D 2 uuxk dxk for k D 1; 2 ; 1

so that max u2 .x1 ; x2 / 2

Z

xk

and by the Schwarz inequality, Z Z 4 u dx R2

1 1

juuxk j dxk

C1

max u dx1 x2

1

Z 4

C1 1

(3.9)

max u2 dx2 x1

Z R2

Z 2

Z

2

for k D 1; 2 ;

R2

juux2 jdx juj2 dx

Z R2

R2

juux1 jdx

jDuj2 dx;

which proves the lemma for smooth functions with compact support in ˝.

t u

Lemma 3.8 (Ladyzhenskaya Inequality in Three Dimensions). Let ˝ R3 be an open and bounded set. Then for all u 2 H01 .˝/ kukL4 .˝/

p 1=4 3=4 2kukL2 .˝/ kDukL2 .˝/ :

(3.10)

Proof (cf. [146]). As in the above lemma, it suffices to prove inequality (3.10) for smooth functions with compact support. Let u 2 C01 .˝/. We have, by (3.8) and (3.9), Z Z Z C1 Z u4 dx 2 u2 dx1 dx2 .u2x1 C u2x2 /dx1 dx2 dx3 R3

R2

1

Z 2 max x3

R2

u2 dx1 dx2

R2

Z R3

jDuj2 dx

52


Z 4

Z juux3 jdx

R3

Z 4

R3

juj2 dx

R3

jDuj2 dx

1=2 Z R3

jDuj2 dx

3=2

; t u

which gives inequality (3.10). 3

Lemma 3.9 (cf. [146]). Let ˝ R be an open and bounded set. Then for all u 2 H01 .˝/, kukL6 .˝/ 481=6 kDukL2 .˝/ :

(3.11)

Proof. Without loss of generality we can assume that u 2 C01 .˝/ and u 0. We have Z Z C1 Z I u6 dx D u3 u3 dx2 dx3 dx1 R3

Z

C1

Z

C1

1

Z 9

R2

ju ux2 jdx2 dx3 Z

u dx2 dx3

R2

3

max u dx2 dx1 x3

1

Z

4

R2

C1

2

(Z R1

x2

Z

1

Z

3

max u dx3

1

C1

Z 9

R2

1

2

R2

ju ux3 jdx2 dx3 dx1

u2x2 dx2 dx3

1=2 Z

u2x3 dx2 dx3

R2

1=2 )

Now we use the Schwarz inequality and proceed as follows: Z I 9 max x1

Z 36

R2

ju ux1 jdx

3 p Y 36 I iD1

R3

Z

3

R3

Z

u4 dx2 dx3

Z R3

R3

ju2xi jdx

u2x2 dx

ju2x2 jdx 1=2

:

1=2 Z R3

1=2 Z R3

u2x3 dx

ju2x3 jdx

1=2

1=2

dx1 :


53

In the end, ( 3 ) 1=2 3 Y Y p I 36 kuxi kL2 .˝/ D 36 kuxi k2L2 .˝/ iD1

8 <

iD1

36 33 :

3 X

kuxi k2L2 .˝/

iD1

!3 91=2 = ;

(as .a1 a2 a3 /1=3 .a1 C a2 C a3 /=3 for ai 0), which gives inequality (3.11).

t u

Theorem 3.12 (Hardy Inequality, cf. [112, 142]). Let a; b 2 R, a < b, u 2 Lp .a; b/ with p > 1. Then Z

b

a

1 xa

Z

x

p ju.s/jds

dx

a

p p1

p Z

b

ju.x/jp dx;

(3.12)

ju.x/jp dx:

(3.13)

a

and Z

b

a

1 bx

Z

b

p ju.s/jds

dx

x

p p1

p Z

b

a

Proof. Let un .x/ D 0 for x 2 a; a C 1n and un .x/ D u.x/ for x 2 a C 1n ; b . We prove the first inequality. Integration by parts and Hölder inequality give Z

b

a

1 xa

Z

x

p jun .s/jds

Z

1

dx D

b

p jun .s/jds

.b a/p1 .1 p/ a Z x .p1/ Z b 1 p C jun .s/jds jun .x/jdx p 1 a .x a/p1 a Z x p1 Z b 1 p ju.s/jds jun .x/jdx p 1 a .x a/p1 a Z b Z x p .p1/=p Z b 1=p p 1 p ju .s/jds dx ju .x/j dx : n n p 1 a .x a/p a a a

Thus, Z

b a

1 xa

Z

x a

p jun .s/jds

dx

p p1

p Z

b

ju.x/jp dx

(3.14)

a

for n 2 N. Applying the Fatou lemma we obtain (3.12). In a similar way we obtain the second inequality.

54


Exercise 3.11. Deduce inequality (3.12) from inequality (3.14) and prove inequality (3.13). Exercise 3.12. Prove that for v 2 W 1;p .a; b/ with v.a/ D 0, p > 1, the following inequality holds: Z a

b

ˇ ˇ p Z ˇ v.x/ ˇp p ˇ dx ˇ ˇx aˇ p1

b

jv 0 .x/jp dx:

a

3.4 Sobolev Spaces of Periodic Functions We consider Sobolev spaces Hps .Q/ of L-periodic functions on the m-dimensional domain Q D Œ0; Lm . Let Cp1 .Q/ be the space of restrictions to Q of infinitely differentiable functions which are L-periodic in each direction, that is, u.x C Lej / D u.x/, j D 1; : : : ; m (by ej we denote the vectors of the canonical basis, ej D .0; : : : ; 1; : : : ; 0/, where 1 is on the jth coordinate). Definition 3.11. For an arbitrary nonnegative integer s we define the Sobolev space Hps .Q/ as the completion of Cp1 .Q/ in the norm 8 91=2 < X = kukHps .Q/ D kD˛ uk2L2 .Q/ : : ; 0j˛js

We denote, ˛ D .˛1 ; : : : ; ˛m / for nonnegative integers ˛1 ; : : : ; ˛m , j˛j D ˛1 C C j˛j ˛m ˛m , ˛Š D ˛1 Š˛2 Š ˛m Š. For x D .x1 ; : : : ; xm /, x˛ D x1˛1 x2˛2 xm , D˛ D @@x˛ . In this notation the Taylor series in Rm can be written just as f .x/ D

X 1 D˛ f .x0 /.x x0 /˛ : ˛Š

j˛j0

Characterization of Sobolev Spaces Hps .Q/ by Using the Fourier Series Each function u 2 Cp1 .Q/ has a representation u.x/ D

X

x

ck e2ik L ;

k2Zm

where ck are complex numbers. We consider real functions, for which ck D ck . It is known that for functions from Cp1 .Q/ the above series converges uniformly. We have 2i j˛j X x ck k˛ e2ik L : D˛ u.x/ D L k2Zm

3.4 Sobolev Spaces of Periodic Functions

55

From the Parseval identity kD˛ uk2L2 .Q/ D Lm

2 L

2j˛j X

jck j2 k2˛ :

k2Zm

We introduce a new norm in Hps .Q/ by ( kukHfs .Q/ D

X 1 C jkj2s jck j2

) 1=2 :

k2Zm

Lemma 3.10. The norms k kHps and k kHfs .Q/ are equivalent, that is c1 kukHfs .Q/ kukHps .Q/ c2 kukHfs .Q/ for all u 2 Hps .Q/ and some positive constants c1 ; c2 . Proof. For every integer s there exist positive constants C1 , C2 such that for all k 2 Rm P 2˛ 0j˛js k C1 C2 : (3.15) 1 C jkj2s Now, X

kuk2Hps .Q/ D

kD˛ uk2L2 .Q/ D Lm

0j˛js

c

X k2Zm

0 @

X

X 2 2j˛j L

0j˛js

1

jkj2j˛j A jck j2 cC2

0j˛js

X

! jck j2 k2˛

k2Zm

X 1 C jkj2s jck j2 D cC2 kuk2H s .Q/ : f

k2Zm

Setting c2 D cC2 we have the second inequality of the lemma. Using once again (3.15) we obtain the first inequality of the lemma. t u We have thus Lemma 3.11. The space Hps .Q/ coincides with ( u 2 L2 .Q/ W u.x/ D

X

2ik Lx

ck e

;

ck D ck ;

k2Zm

X

) jkj2s jck j2 < 1 :

k2Zm

Lemma 3.12. In the space Z P ps .Q/ D u 2 Hps .Q/ W H u.x/dx D 0 Q

56


(of these u 2 Hps .Q/ for which c0 D 0 in their series representation) the expression X

kukHP ps .Q/ D

!1=2 2s

jkj jck j

2

k2Zm

is a norm equivalent to the norm k kHps .Q/ . Proof. The proof follows from the Poincaré inequality L kDukL2 .Q/ ; 2

kukL2 .Q/

(3.16)

P p1 .Q/. In fact, we have which holds for all u in H X

kDuk2L2 .Q/ D

kD˛ uk2L2 .Q/ D

j˛jD1

X

Lm

j˛jD1

2 L

2 L

X

m

2

2 L

X

jck j2 jkj2

k2Zm nf0g

2

jck j D

k2Zm nf0g

2 L

2

kukL2 .Q/ ; t u

which ends the proof. In the space k kHP ps .Q/ we introduce the scalar product .u; v/HP ps .Q/ D

X

jkj2s ck dk ;

k2Zm

for u.x/ D

P

x

k2Zm

ck e2ik L and v.x/ D ck D uO .k/ D

P

1 Lm

x

k2Zm

Z

dk e2ik L . We have also x

u.x/e2ik L dx: Q x

The Fourier coefficients uO of u are complex amplitudes of harmonics e2ik L associated to wave numbers 2k . The length scale related to the wave number 2k L L L equals jkj . Notice that L L L L Dq : p jkj kmin mkmax k12 C C km2 Theorem 3.13 (Sobolev Embedding Theorem). Let u 2 Hps .Q/, s > u 2 Cp0 .Q/ and sup ju.x/j D kukL1 .Q/ C.s/kukHps .Q/ : x2Q

m . 2

Then


Proof. Let u.x/ D kukL1 .Q/

P

X (

x

k2Zm

jck j

k2Zm

X k2Zm

57

ck e2ik L . Then, for s > X k2Zm

1 1 C jkj2s

1 jkj2s /1=2

.1 C ) 1=2 (

m 2

we have

1=2 1 C jkj2s jck j

X 1 C jkj2s jck j2

) 1=2 D C.s/kukHps .Q/ :

k2Zm

P Moreover, the absolute convergence of the series of coefficients k2Zm jck j implies the uniform convergence of the Fourier series of u and, in consequence, the continuity of u. t u P 1 Exercise 3.13. Prove that for s > m2 the series k2Zm 1Cjkj 2s is convergent. Exercise 3.14. Prove that if s > m2 D j and u 2 Hps .Q/ then u is j times continuously differentiable function in Q (we write u 2 Cj .Q/) and X sup jD˛ u.x/j C.s/kukHps .Q/ : kukCj .Q/ D j˛jj

x2Q

We shall need also the Rellich–Kondrachov theorem on compact embedding. Theorem 3.14 (Rellich–Kondrachov). The space Hp1 .Q/ is compactly embedded in the space L2 .Q/ which means that for every bounded in Hp1 .Q/ sequence of functions there exists its subsequence that is convergent in L2 .Q/. P x Proof. Let fun g, un .x/ D k2Zm cn;k e2ik L , n 2 N be a bounded sequence in Hp1 .Q/, that is, X 1 C jkj2 jcn;k j2 M k2Zm

for all n and some positive constant M. We have P to prove x that there exists a subsequence uj which converges to some u? D k2Zm c?k e2ik L 2 Hp1 .Q/ strongly in L2 .Q/. It is known that from every bounded sequence in a Hilbert space one can choose a weakly convergent subsequence. Assume that uj converges weakly to u? in Hp1 .Q/. Then u? satisfies X 1 C jkj2 jc?k j2 M: k2Zm

Exercise 3.15. Let the sequence uj in a Hilbert space H converge weakly to u? 2 H, that is, for every v 2 H, .uj u? ; v/H ! 0 as j ! 1, where .; /H denotes the scalar product in H. Prove that the norm of u? in H satisfies ku? kH lim inf kuj kH : j!1

58


Exercise 3.16. Let uj converge weakly to u? in Hp1 .Q/. Prove that it is equivalent to the convergence of all Fourier coefficients, namely, cj;k ! c?k as j ! 1, for every k 2 Zm . We shall prove that the subsequence uj converges to u? in L2 .Q/. Observe that X 1 C jkj2 jcj;k c?k j2 4M; k2Zm

whence kuj u? k2L2 .Q/ D

X

jcj;k c?k j2

X

jcj;k c?k j2 C

jkjK

k2Zm

X

1 X jcj;k c?k j2 jkj2 K2 jkj>K

jcj;k c?k j2 C

jkjK

4M : K2

< 2" . Then, in view of For every " > 0 we can take K large enough so that 4M K2 Exercise 3.16 we can take j large enough so that the first term on the right-hand side is smaller than 2" . This proves the convergence uj ! u? in L2 .Q/. t u Exercise 3.17. Prove that H sC1 .Q/ is compactly embedded in H s .Q/. Laplace Equation in Sobolev Spaces of Periodic Functions Let f 2 L2 .Q/. Assume that we look for u 2 Hps .Q/ satisfying equation u D f :

(3.17)

If u 2 Hps .Q/ is a solution of (3.17) then for an arbitrary constant c, u C c is also a solution in the same space. To guarantee uniqueness we restrict our attention to solutions satisfying the additional property Z u.x/dx D 0;

(3.18)

Q

P ps .Q/. But then we have also which is equivalent to u 2 H Z u.x/dx D 0:

(3.19)

Q

Exercise 3.18. Prove that if u 2 Hps .Q/ satisfies (3.18) then also (3.19) holds. It then follows that Z f .x/dx D 0; Q


59

P p0 .Q/. For any f 2 L2 .Q/ there exists a constant c such that whence f is in LP 2 .Q/ D H f C c 2 LP 2 .Q/. In consequence, to have the solution uniqueness, we shall assume that f 2 LP 2 .Q/. Let X x fk e2ik L ; f0 D 0 f .x/ D k2Zm

and u.x/ D

X

x

ck e2ik L ;

u0 D 0:

(3.20)

k2Zm

We compute u.x/ D

2 L

2 X

X

x

jkj2 uk e2ik L D

k2Zm

x

fk e2ik L D f ;

k2Zm

and comparing the coefficients we obtain uk D

L 2 fk 4 2 jkj2

k 6D 0:

for

(3.21)

We can prove now the following: P p1 .Q/ are a generalized solution of the Lemma 3.13. If f 2 LP 2 .Q/ and u 2 H Laplace equation, that is, u satisfies the integral identity Z Z ru W rv dx D f v dx (3.22) Q

Q

P p2 .Q/ and for all v in CP p1 .Q/, then u 2 H kukHP p2 .Q/ Ckf kL2 .Q/ :

(3.23)

Proof. First, observe that in the above integral identity we can take the test functions P p1 .Q/. Then the existence of a unique generalized solution in from the space H P p1 .Q/ follows easily from the Riesz–Frèchet theorem. This solution is given H by (3.20) where the coefficients are given in (3.21). Moreover, the solution belongs P p2 .Q/, as to H kuk2HP 2 .Q/ D p

X

jkj4 juk j2 D

k2Zm

k2Zm

D This ends the proof.

X

jkj4

L 2 fk 4 2 jkj2

2

L4 X L4 2 jf j D kf kL2 .Q/ : k 16 4 k2Zm 16 4 t u

60


Looking on the equation u D f as on the abstract operator equation Au D f in the space LP 2 .Q/ with operator A W LP 2 .Q/ D.A/ ! LP 2 .Q/ we see that P p2 .Q/: P p1 .Q/ W Au 2 LP 2 .Q/g D H D.A/ D fu 2 H P p2 .Q/, on the other hand, for every u in In fact, from (3.23) it follows that D.A/ H 2 2 2 P p .Q/ D.A/. P p .Q/, Au 2 LP .Q/, whence H H P p1 .Q/ functions f 2 H P ps .Q/ and u 2 H P p1 .Q/ Exercise 3.19. Prove that if for all v 2 H satisfy (3.22) then u 2 HpsC2 .Q/ and (3.23) holds. Lemma 3.14. The operator A is invertible on LP 2 .Q/ and A1 W LP 2 .Q/ ! LP 2 .Q/ is compact, that is, it maps bounded sets in LP 2 .Q/ to precompact sets in LP 2 .Q/. Proof. From the above considerations we know that the operator A1 maps LP 2 .Q/ P p2 .Q/ and that the correspondence is one to one. From Theorem 3.14 it follows onto H P p2 .Q/ is compactly embedded in LP 2 .Q/, whence A1 is compact. t u that H Lemma 3.15. In the considered case of periodic boundary conditions the eigenfunctions of the Laplacian operator in LP 2 .Q/ belong to CP p1 .Q/. P p2 .Q/. Thus Proof. Let Aw D w 2 LP 2 .Q/. From Lemma 3.13 it follows that w 2 H P p4 .Q/. P p2 .Q/. Making use of Exercise 3.19 we conclude that w 2 H Aw D w 2 H s P p .Q/ for each integer s. Now, it suffices to use the By induction we obtain w 2 H t u Sobolev embedding theorem to conclude that w belongs to CP p1 .Q/. P p1 .Q/, and Au D f then u is infinitely Exercise 3.20. Prove that if f 2 CP p1 .Q/, u 2 H 1 smooth, namely, u 2 CP p .Q/. Theorem 3.15 (cf. [197, Corollary 3.26]). Let H be a Hilbert space and let A be a symmetric linear operator in H whose range is all of H, and suppose further that its inverse is defined and compact. Then A has an infinite set of real eigenvalues n with corresponding eigenfunctions !n : A!n D n !n . If the eigenvalues are ordered so that jnC1 j jn j one has limn!1 jn j D 1. Furthermore, the !n can be chosen so that they form an orthonormal basis for H, and in terms of this basis the operator A can be represented by Au D

1 X

j .u; !j /H !j

jD1

on its domain

8 9 1 1 < = X X D.A/ D u W u D cj !j ; jcj j2 2j < 1 : : ; jD1

jD1

D.A/ is a Hilbert space with the inner product .u; v/D.A/ D .Au; Av/H and corresponding norm kukD.A/ D kAukH .

3.5 Evolution Spaces and Their Useful Properties

61

For the operator Au D u in LP 2 .Q/, whence we have

X X x x x C i sin 2k ck e2ik L D ck cos 2k u.x/ D L L m m k2Z nf0g

k2Z nf0g

r D

where

r .c/ !k .x/ .:/

and A!k D

D

Lm 2

X

.c/

.s/

Re.ck /!k Im.ck /!k

;

k2.Zm nf0g/=2

2 x ; cos 2k Lm L

4 2 jkj2 .:/ !k , L2

r .s/ !k .x/

D

2 x ; sin 2k Lm L

.:/

k!k kL2 .Q/ D 1. 2

jkj2 with jkj D For example, for m D 2, the smallest eigenvalue is 4 L2 It repeats four times, 1 D 2 D 3 D 4 in the sequence

q k12 C k22 D 1.

0 < 1 2 3 : : : n : : : : .c/

.s/

The corresponding eigenfunctions are given as !1 D !.1;0/ , !2 D !.1;0/ , !3 D .c/

.s/

!.0;1/ , !4 D !.0;1/ .

3.5 Evolution Spaces and Their Useful Properties The solutions of evolution problems are the functions which depend on both space and time variables. Typically, the space variable belongs to a certain domain ˝ Rd and the time variable belongs to the time interval I D Œt0 ; t1 . A typical technique is to deal with these variables differently. Namely, assume that u is a function of .x; t/ 2 ˝ I and freeze t 2 I. Then u.; t/ is only a function of the space variable. Usually we assume that this function belongs to a certain Banach space of functions defined on ˝, let us denote this space by V. We write u.t/ 2 V. If we “unfreeze” the time variable now, we can write u W I ! V, so the function of .x; t/ 2 ˝ I is treated as the Banach space valued function of a time variable only. This chapter is devoted to the recollection of the definition and basic properties of spaces of such functions. Definition 3.12. Let V be a separable Banach space. The function u W I ! V is strongly measurable if there exists a sequence of simple functions un W I ! V such that lim un .t/ D u.t/

n!1

for a.e.

t 2 I:

P A function v W I ! V is simple if v.t/ D m kD1 Ak .t/wk for all t 2 I, where Ak are Lebesgue measurable subsets of I, and wk 2 V.

62


Definition 3.13. Let V be a separable Banach space. A strongly measurable function v W I ! V is Bochner integrable if Z kv.t/kV dt < 1: I

The space of Bochner integrable functions is denoted by L1 .II V/. It is possible to use the equivalent definition of this space, which gives rise to the so-called Bochner integral. Theorem 3.16 (cf. [85, Theorem 2, p. 45]). Let v W I ! V be a strongly measurable function. Then v 2 L1 .II V/ is Bochner integrable if and only if there exists a sequence of simple functions vn W I ! V such that Z lim

n!1 I

kvn .t/ v.t/kV dt D 0:

(3.24)

Definition 3.14. If the function v W I ! V is simple then its Bochner integral over the Lebesgue measurable set E I can be defined as Z v.t/ dt D E

m X

wk m.Ak \ E/:

iD1

If, in turn, v 2 L1 .II V/, then the Bochner integral is defined as Z

Z v.t/ dt D lim

n!1 E

E

vn .t/ dt;

where vn is the sequence of simple functions from Theorem 3.16. It is easy to prove that the value of the Bochner integral does not depend on the choice of the sequence of simple functions which satisfy (3.24). More information on Bochner integral and its properties can be found in [85, 107, 234]. The definitions of Bochner integral and the space L1 .II V/ do not require the separability of V. In such a case, however, the notion of measurability of a Banach space valued function should be defined more carefully. For the sake of simplicity, however, in this book we always use this notions for separable spaces V. As a natural extension of L1 .II V/ it is possible to define the spaces Lp .II V/ for p 2 Œ1; 1 as the spaces of the measurable functions, such that ( R kf kLp .IIV/ D

I

p

kf .t/kV dt

1p

<1

for

ess supt2I kf .t/kV < 1 for

p 2 Œ1; 1/; p D 1:

Once we identify any two functions which differ only of a subset of I of Lebesgue measure zero, the space Lp .II V/ endowed with the norm k kLp .IIV/ becomes a


63

Banach space. Moreover, through a natural isometric isomorphism we can make the following identification Lp .II Lp .˝/m / D Lp .˝ I/m valid for p ¤ 1 (note that L1 .˝/m is not a separable space, see Example 1.42 in [204] and Example 23.4 in [234]). We have the following useful results related to the dual space to Lp .II V/. Proposition 3.1 (cf. [234, Proposition 23.7]). Let V be a separable and reflexive Banach space and let 1 < p < 1, and 1p C 1q D 1. Then Lp .II V/ is a reflexive and separable Banach space. Moreover, the dual space of Lp .II V/ is isometrically isomorphic with Lq .II V 0 /. Proposition 3.2 (cf. [234, Proposition 23.9]). Let V be a separable and reflexive Banach space and let 1 < p < 1, and 1p C 1q D 1. If u 2 Lp .II V/ and v 2 V 0 , then

Z v; u.t/ dt I

Z D

hv; u.t/iV 0 V dt:

V 0 V

I

Moreover from un ! u

strongly in Lp .II V/;

vn ! v

weakly in

Lq .II V 0 /;

it follows that Z

Z hvn .t/; un .t/iV 0 V dt !

hv.t/; u.t/iV 0 V dt:

I

I

A function v 2 L1 .II V/ is called a distributional time derivative of a function u 2 L1 .II V/, provided the following equality Z Z u.t/ 0 .t/ dt D v.t/.t/ dt for all 2 C01 .I/; (3.25) I

I

holds in V. In such a case we write v D u0 D ut D du . Sometimes, the distributional dt time derivative of a function belonging to L1 .II V/ may not exist as a function belonging to the same space, but it may have values in a larger Banach space Z, such that we have the embedding V Z. Then, the function v 2 L1 .II Z/ is a distributional time derivative of u 2 L1 .II V/, provided (3.25) holds in V. Note that, in such a case, the integral on the right-hand side of (3.25) belongs to V, although the integrand has values only in Z, and hence not necessarily in V. As it can be easily checked, the distributional derivative, provided it exists as a function u0 2 L1 .II Z/, is unique with respect to equality for a.e. t 2 I. Moreover we have the following equality in Z valid for all s; t 2 I Z u.t/ D u.s/ C s

t

u0 .r/ dr:

64


Note that if u 2 L1 .II V/ and u0 2 L1 .II Z/ with V Z, then, after modification of the null set we must have u 2 C.II Z/ and hence the pointwise values of u make sense as the elements of Z in the last equality. An important role in the study of weak solutions of evolution problems is played by the so-called evolution triples (or Gelfand triples). An evolution triple consists of the reflexive Banach space V (we will always assume that V is also separable), the Hilbert space H such that we have the continuous and dense embedding V H, and the dual space V 0 . As the space H is always identified with its dual we can write V H V 0 with all embeddings being continuous and dense. From now on we will always denote the scalar product on H by .; / and associated norm by j j. The norm on V is denoted by k k and the duality pairing between V and V 0 by h; i: An important property of evolution triples is the following formula hu; vi D .u; v/

whenever

u2H

v 2 V:

and

(3.26)

The following results which hold in the framework of evolution triples will be frequently used later. Corollary 3.2. Let V H V 0 be an evolution triple such that V H is additionally compact and V is Hilbert. Let A W V ! V 0 be a linear operator which is • bounded (kAukV 0 kAkL .VIV 0 / kuk for all u 2 V), • symmetric (hAu; vi D hAv; ui for all u; v 2 V), • coercive (hAu; ui ˛kuk2 for all u 2 V with ˛ > 0). Then A satisfies the assumptions of Theorem 3.15 and we can construct an orthogonal in V and orthonormal in H basis of eigenfunctions !j of A with the corresponding eigenvalues such that jj j ! 1 as j ! 1. Proof. The Lax–Milgram lemma (cf. Lemma 3.2) implies that for any f 2 H there exists a unique v 2 V such that Av D f . Hence the range of A (treated as an operator from D.A/ H to H) is whole H and A1 W H ! H is well defined. Moreover, by (3.26), A, when treated as an operator on H, is symmetric. Obviously A1 is linear. To prove compactness of A1 let fn be a bounded sequence in H. Let un 2 V be such that Aun D fn . Coercivity of A implies that the sequence un is bounded in V and hence, by the compactness of the embedding V H, for a subsequence, un ! u strongly in H for certain u 2 H. The proof of compactness is complete. t u Theorem 3.17 (cf. [197, Lemma 7.5]). Let V H V 0 , A W V ! V 0 , and !j be as in Corollary 3.2. Denote Vn D spanf!1 ; : : : ; !n g. If we define Pn W V 0 ! Vn by Pn Pn v D iD1 hv; !i iwi for v 2 V 0 then we have if v 2 V

then

kPn vk kvk

if v 2 H

then

jPn vj jvj and

if v 2 V 0

then

kPn vkV 0 kvkV 0

and

lim kPn v vk D 0;

n!1

lim jPn v vj D 0;

n!1

and

We will frequently use the following two results.

lim kPn v vkV 0 D 0:

n!1


65

Lemma 3.16 (cf. [234, Proposition 23.23], [219, Chap. 3, Lemma 1.2]). Let V H V 0 be an evolution triple, let I D .t1 ; t2 / be a finite time interval, and let 1 < p < 1 and 1p C 1q D 1. Then 1. The space W .I/ D fu 2 Lp .II V/ W u0 2 Lq .II V 0 /g is a Banach space equipped with the norm kukW .I/ D kukLp .IIV/ C ku0 kLq .IIV 0 / : 2. The space W .I/ embeds continuously in C.II H/, i.e., every function from W .I/ is almost everywhere equal to a continuous function with values in H, and we have max ju.t/j CkukW .I/

u 2 W .I/:

for all

t2I

3. We have the following integration by parts formula valid for all u; v 2 W .I/, Z .u.t1 /; v.t1 // .u.t0 /; v.t0 // D hv 0 .t/; u.t/i C hu0 .t/; v.t/i dt: I

4. For any u 2 W .I/ the following equality holds in the sense of distributions on I d ju.t/j2 D 2hu0 .t/; u.t/i: dt Corollary 3.3. Let V; H; p; q be as in the previous Lemma. Let un ! u weakly in Lp .II V/ and u0n ! v weakly in Lq .II V 0 /. Then v D u0 and un .t/ ! u.t/ weakly in H for all t 2 I. Proof. The fact that v D u0 follows, for example, from Proposition 23.19 in [234]. We will prove the second assertion. We have Z un .s/ D un .t/ C s

t

u0n .r/dr

s; t 2 I:

for all

Taking the duality with any v 2 V and integrating with respect to t over I we get Z .t1 t0 /.un .s/; v/ D

Z Z .un .t/; v/ dt C

I

I

Z

Z

.un .t/; v/ dt C

D I

I

s

t

hu0n .r/; vi dr dt

hu0n .t/; vs .t/i dt;

where s D .t0 s/.t0 ;s/ C .t1 s/.s;t1 / . The assertion follows by passing to the limit n ! 1 in the last formula and using the density of the embedding V H. u t We now recall two important results on the triples of spaces V1 V2 V3 , where the embedding V1 V2 is compact. The first result is known as the Ehrling lemma.

66


Lemma 3.17 (See [204, Lemma 7.6]). Let V1 V2 V3 be three Banach spaces with all embeddings continuous and V1 V2 compact. Then for any " > 0 there exists a constant C."/ such that kvkV2 "kvkV1 C C."/kvkV3

v 2 V1 :

for all

The next result is known as Aubin–Lions compactness theorem. Theorem 3.18 (cf. [6, 155], [216, Corollary 4], [204, Lemma 7.7], [219, Chap. 3, Theorem 2.1]). Let I be a bounded time interval and let 1 p; q < 1 and let X1 X2 X3 be three separable Banach spaces with all embeddings being continuous. If the embedding X1 X2 is compact, then the embedding fu 2 Lp .II X1 / W u0 2 Lq .II X3 /g Lp .II X2 / is also compact. Fairly general form of the above Lemma is given in [216]. Note that the assumption of separability is not required there. A function u W I ! V, where V is a Banach space is said to be weakly continuous provided the function I 3 t ! hv; u.t/i 2 R is continuous for all v 2 V 0 . We write u 2 Cw .II V/, or u 2 C.II Vweak /, or, sometimes C.II Vw /. Of course C.II V/ Cw .II V/ with the strict inclusion. We have the following result. Lemma 3.18 (cf. [219, Chap. 3, Lemma 1.4]). Let X Y be two Banach spaces with a continuous embedding and let I be a finite time interval. If u 2 L1 .II X/ and u 2 Cw .II Y/ then u 2 Cw .II X/. Finally we give a parabolic embedding result. Lemma 3.19 (see [155, Chap. 1, Lemma 6.7]). Let ˝ R3 be an open and bounded set with Lipschitz boundary and let I be a bounded time interval. If u 2 L2 .II H 1 .˝// \ L1 .II L2 .˝//; then u 2 L4 .II L3 .˝//. Proof. By the Sobolev embedding theorem we have H 1 .˝/ L6 .˝/ with a continuous embedding, so, using the Hölder inequality we have for a.e. t 2 I and for a constant c > 0 1

1

1

1

ku.t/kL3 .˝/ ku.t/kL22 .˝/ ku.t/kL26 .˝/ cku.t/kL22 .˝/ ku.t/kH2 1 .˝/ :

3.6 Gronwall Type Inequalities

Thus we have Z I

67

ku.t/k4L3 .˝/ dt c4 ess supfku.t/k2L2 .˝/ g

Z

t2I

I

ku.t/k2H 1 .˝/ dt; t u

and the assertion is proved.

3.6 Gronwall Type Inequalities In this section we deal with Gronwall type inequalities, which are the efficient and useful tool in the study of evolution problems. These inequalities are used in the derivation of the a priori estimates that are satisfied by the solution of the analyzed problem. The first Gronwall type inequality is useful in the study of existence and uniqueness of solutions to initial and boundary value problems for evolutionary PDEs, as well as in the derivation of dissipative a priori estimates for their solutions. 1 Lemma 3.20 (Gronwall Inequality). Let t0 be a real number. Let x 2 Lloc .t0 ; 1/ 1 1 0 be a function such that x 2 Lloc .t0 ; 1//, and let h 2 Lloc .t0 ; 1/ and k 2 1 Lloc .Œt0 ; 1//. If

x0 .t/ x.t/k.t/ C h.t/

for a:e:

t 2 .t0 ; 1/;

(3.27)

then Rt

x.t/ x.t0 /e

t0

Z k.s/ ds

C

Rt

t

h.s/e

s

k.r/ dr

ds

for all

t 2 Œt0 ; 1/:

(3.28)

t0

If, in particular, k.t/ D C for all t t0 , C being a real constant, then x.t/ x.t0 /eCt0 eCt C eCt

Z

t

h.s/eCs ds

for all

t 2 Œt0 ; 1/:

(3.29)

t0

If, furthermore, h.t/ D D for all t t0 , D being a real constant, then x.t/ x.t0 /eCt0 eCt C

D Ct0 Ct .e e 1/ C

for all t 2 Œt0 ; 1/:

Proof. Multiplying (3.27) by the integrating factor e x0 .t/e

Rt t0

k.s/ ds

x.t/k.t/e

Rt t0

k.s/ ds

h.t/e

Rt t0

Rt t0

k.s/ ds

k.s/ ds

we get a.e. t > t0 ;

(3.30)

68


whence Rt Rt d x.t/e t0 k.s/ ds h.t/e t0 k.s/ ds dt

for a.e.

t > t0 :

Integrating the above inequality from t0 to t we obtain (3.28). Assertions (3.29) and (3.30) follow from (3.28) by a straightforward calculation. t u If, in (3.29) or (3.30), we have C < 0, then the resultant estimate is, after appropriately large time, independent on the initial condition x.t0 /. Such estimates are called dissipative and they are particularly useful in the study of global attractors. If, however, we only have (3.27) with k.t/ positive on .t0 ; 1/, then the resultant bounds grow to infinity as t ! 1. In such situation, we need the following uniform Gronwall inequality. 1 Lemma 3.21 (Uniform Gronwall Inequality). Let t0 2 R. Let x 2 Lloc .t0 ; 1/ be 1 1 1 0 a function such that x 2 Lloc .t0 ; 1/, and let h 2 Lloc .t0 ; 1/ and k 2 Lloc .Œt0 ; 1//. Let moreover x.t/ 0, h.t/ 0, and k.t/ 0 for a.e. t > t0 . If

x0 .t/ x.t/k.t/ C h.t/ and Z

tCr t

Z

tCr

k.s/ ds a1 ;

for a:e:

Z

tCr

h.s/ ds a2 ;

t

t 2 .t0 ; 1/;

x.s/ ds a3

for all

(3.31)

t 2 Œt0 ; 1/;

t

where a1 ; a2 ; a3 ; r are positive constants, then x.t C r/

a

3

r

C a2 ea1

t 2 Œt0 ; 1/:

for all

Proof. Let t0 t R p t C r. We multiply (3.31) written for p 2 .t; t C r/ by the integrating factor e t k. / d , getting Rp Rp Rp d x.p/e t k. / d D x0 .p/e t k. / d x.p/k.p/e t k. / d dp h.p/e

Rp t

k. / d

h.p/

a.e.

p 2 .t; t C r/:

We fix s 2 .t; t C r/ and integrate the last inequality over the interval .s; t C r/. We get Z tCr R tCr Rs t k. / d t k. / d x.s/e h.p/ dp a2 ; x.t C r/e s

whence we have the inequality R tCr

x.t C r/ x.s/e

s

k. / d

R tCr

C a2 e

t

k. / d

x.s/ea1 C a2 ea1 ;

3.6 Gronwall Type Inequalities

69

valid for all s 2 .t; t C r/. We integrate the last inequality with respect to s over the interval .t; t C r/, which gives us Z rx.t C r/

tCr

x.s/ ds ea1 C ra2 ea1 a3 ea1 C ra2 ea1 ;

t

t u

which immediately yields the assertion.

If we need to estimate the norm of the unknown function x by a constant that is arbitrarily small, then the following translated Gronwall inequality becomes useful (see [161]). 1 Lemma 3.22. Let > 0 be a constant. Let x 2 Lloc .t0 ; 1/ be such that x0 2 1 1 Lloc .t0 ; 1/ and x.t/ 0 for a.e. t t0 . Let moreover h 2 Lloc .t0 ; 1/ be such that h.t/ 0 for a.e. t t0 . If we have

x0 .t/ C x.t/ h.t/

t > t0 ;

a:e:

(3.32)

then for all t t0 and r > 0 we have x.t C r/

Z

2 r e 2 r

tC 2r

x.s/ ds C e.rCt/

Z

t

tCr

h.s/es ds:

t

In particular, taking r D 2, we get, for all t t0 x.t C 2/ e

Z

tC1

x.s/ ds C e.tC2/

t

Z

tC2

h.s/es ds:

t

Proof. Multiplying (3.32) written for p t0 by the integrating factor ep , we get d .x.p/ep / D x0 .p/ep C x.p/ep h.p/ep dp

a.e.

p t0 :

Let t t0 and r > 0. We fix s 2 t; t C 2r and integrate the last inequality over the interval .s; t C r/ leading to x.t C r/e.tCr/ x.s/es

Z

tCr

h.p/ep dp;

s

whence we have .str/

x.t C r/ x.s/e

.tCr/

Z

Ce

tCr

h.p/ep dp

s

.str/

x.s/e

.tCr/

Z

Ce

t

tCr

h.p/ep dp:

70


We integrate the last inequality with respect to s over the interval .t; t C 2r /, which gives r x.t C r/ 2

Z

tC 2r

.str/

x.s/e

t

r ds C e.tCr/ 2

Z

tCr

h.s/es ds;

t

t u

and the assertion follows immediately. We shall use also the following lemma.

Lemma 3.23 (Generalized Gronwall Inequality). Let ˛ D ˛.t/ and ˇ D ˇ.t/ be locally integrable real-valued functions on Œ0; 1/ that satisfy the following conditions for some T > 0: lim inf t!1

lim sup t!1

1 T

1 t!1 T

1 T Z Z

Z

tCT

˛./d > 0;

(3.33)

˛ . /d < 1;

(3.34)

ˇ C . /d D 0;

(3.35)

t tCT

t

lim

tCT

t

where ˛ .t/ D maxf˛.t/; 0g, ˇ C .t/ D maxfˇ.t/; 0g. Suppose that D .t/ is an absolutely continuous nonnegative function on Œ0; 1/ that satisfies the following inequality almost everywhere on .0; 1/, d.t/ C ˛.t/.t/ ˇ.t/: dt

(3.36)

Then .t/ ! 0 as t ! 1. For the proof, see [99].

3.7 Clarke Subdifferential and Its Properties We start this section from the recollection of the definition and properties of the Clarke subdifferential, a generalization of Gâteaux derivative to functionals which are only locally Lipschitz. The Clarke subdifferential at a given point assumes multiple values, so it is a multifunction. Then, we give an example of a multivalued partial differential equation, where the semilinear term has the form of a Clarke subdifferential. Finally, we show several examples how to compute Clarke subdifferentials and we discuss the usefulness of the Clarke subdifferential to express various boundary conditions in mechanics. Throughout this chapter X is always assumed to be a Banach space.

3.7 Clarke Subdifferential and Its Properties

71

Definition 3.15. A functional j W X ! R is locally Lipschitz if for every x 2 X there exists an open neighborhood O 2 N .x/ and a constant kO such that for all z; y 2 O jj.z/ j.y/j kO kz ykX . The following two definitions were introduced by Clarke (see [71, 72]). Definition 3.16. Let j W X ! R be locally Lipschitz and let x; h 2 X. A generalized directional derivative in the sense of Clarke of the functional j at the point x 2 X and in the direction h 2 X, denoted by j0 .xI h/ is defined as j0 .xI h/ D lim sup

y!x;!0C

j.y C h/ j.y/ :

Definition 3.17. Let j W X ! R be locally Lipschitz. The generalized subdifferential in the sense of Clarke of the functional j at the point x 2 X is the set @j.x/ X 0 defined as @j.x/ D f 2 X 0 W h; hiX 0 X j0 .xI h/

for all

h 2 Xg:

(3.37)

We recall some results on the properties of the Clarke subdifferential and Clarke generalized directional derivative. Proposition 3.3 (cf. [72, Proposition 2.1.1]). If j W X ! R is locally Lipschitz, then j0 W X X ! R is upper semicontinuous, i.e., n ! x; hn ! h ) lim sup j0 .xn I hn / j0 .xI h/

for all

x; h 2 X:

n!1

Moreover, j0 .xI / is Lipschitz on X with the Lipschitz constant equal to the local Lipschitz constant kO of j at x given in Definition 3.15. Proposition 3.4 (cf. [72, Proposition 2.1.2]). If j W X ! R is locally Lipschitz, then for every x 2 X, the set @j.x/ is nonempty, weakly-* compact, and convex. Moreover, for every 2 @j.x/ we have kkX kO (where kO is the local Lipschitz constant of j at x given in Definition 3.15) and we have j0 .xI h/ D max h; hiX 0 X [email protected]/

for all

x; h 2 X:

Proposition 3.5 (cf. [72, Proposition 2.1.5]). If j W X ! R is locally Lipschitz, then 0 the graph of a multifunction X 3 X ! @j.x/ 2 2X is a strong-weak-* sequentially closed set. For a functional defined on Rd one can calculate its Clarke subdifferential from the following characterization.

72


Theorem 3.19 (see [72, Theorem 2.5.1]). Let j W Rd ! R be locally Lipschitz then @j.s/ D convf lim rj.sn / W sn ! s; sn … S [ Nj g for s 2 Rd ; n!1

(3.38)

where S is any Lebesgue null set, and Nj is the Lebesgue null set on which j is not differentiable. We recall the Lebourg mean value theorem (cf. [72, Theorem 2.3.7]) Theorem 3.20. Let X be a Banach space and j W X ! R be a locally Lipschitz functional. For any x; y 2 X there exists 2 .0; 1/ and 2 @j. x C .1 /y/ such that h; y xiX 0 X D j.y/ j.x/. We are in position to prove the approximation theorem for Clarke subdifferentials. Theorem 3.21. Let j W Rd ! R be a locally Lipschitz functional such that the growth condition jj C.1 C jsj/ holds for all s 2 Rd and all 2 @j.s/ with C > 0. Let moreover n W Rd ! R be a sequence of mollifier kernels given by n .r/ D nd .nr/ Rfor r 2 Rd , where W Rd ! R is a function such that 2 C1 .Rd /, .s/ 0 on Rd , Rd .s/ ds D 1, and supp Œ1; 1d . Define Z n .r/j.s r/ dr: jn .s/ D Rd

Then (A) jn 2 C1 .Rd /, (B) jrjn .s/j D.1 C jsj/ for all s 2 Rd with D > 0 independent of n, (C) if .˝; ˙; / is a finite and complete measure space and vn 2 L2 .˝/d , is a sequence such that vn ! v strongly in L2 .˝/d and rjn .vn / ! weakly in L2 .˝/d , then .x/ 2 @j.v.x// -a.e. in ˝. Proof. The assertion .A/ follows, for example, from [93], Theorem 6, Appendix C. Let us prove .B/. Let h 2 Rd . We have jn .s C h/ jn .s/ D lim !0 !0

rjn .s/ h D lim

Z supp n

n .r/

j.s C h r/ j.s r/ dr:

Using the Lebourg mean value theorem we get ˇ ˇ ˇ j.s C h r/ j.s r/ ˇ ˇ jz.s; h; r/jjhj C.1 C jsj C jrj C jjjhj/jhj; ˇ ˇ ˇ where z.s; h; r/ 2 @j.s r C h/ with 2 .0; 1/ possibly changing from point to point. We get Z jrjn .s/ hj C lim

!0 supp n

n .r/.1 C jsj C jrj C jjjhj/jhj dr:

3.7 Clarke Subdifferential and Its Properties

73

d As supp n 1n ; 1n we have 1 n .r/ 1 C jsj C C jj dr C.2 C jsj/; n supp n

Z jrjn .s/j C lim

!0

and the assertion .B/ follows. We pass to the proof of .C/. As vn ! v in L2 .˝/d we also have, for a subsequence, vn .x/ ! v.x/ and jvn .x/j h.x/ for -a.e. x 2 ˝ with h 2 L2 .˝/. The further argument will be done for this subsequence. Take w 2 L1 .˝/d . From the weak convergence rjn .vn / ! it follows that Z

Z ˝

.x/ w.x/ d .x/ D lim

n!1 ˝

rjn .vn .x// w.x/ d .x/

Z

˝

lim sup rjn .vn .x// w.x/ d .x/; n!1

where the use of the Fatou lemma is justified by the growth condition .B/ and the bound jvn .x/j h.x/. We estimate the integrand for -a.e. x 2 ˝ as follows lim sup rjn .vn .x// w.x/ D lim sup lim n!1

n!1

Z

D lim sup lim

n!1 !0C

lim sup sup

n!1 r2supp n !0C

D lim sup n!1 !0C

!0C

supp n

n .s/

jn .vn .x/ C w.x// jn .vn .x//

j.vn .x/ C w.x/ s/ j.vn .x/ s/ ds

j.vn .x/ C w.x/ r/ j.vn .x/ r/

j.vn .x/ C w.x/ rn; / j.vn .x/ rn; / ;

where rn; 2 supp n is such that the supremum over r of the difference quotient is achieved. Observe that as n ! 1 we must have vn .x/ rn; ! v.x/. From the definition of the Clarke directional derivative we get lim sup rjn .vn .x// w.x/ lim sup n!1

z!v.x/ !0C

j.z C w.x// j.z/ D j0 .v.x/I w.x//;

whereas Z

Z ˝

.x/ w.x/ d .x/

˝

j0 .v.x/I w.x// d .x/:

(3.39)

We will show that .x/ z j0 .v.x/I z/ -a.e. in ˝ for all z 2 Rd and in consequence we will have the assertion .x/ 2 @j.v.x// -a.e. in ˝. First we prove

74


that j0 .v.x/I w.x// .x/ w.x/ 0 for -a.e. x 2 ˝ for all w 2 L1 .˝/d . Indeed, suppose it is not the case. Then for a certain wN 2 L1 .˝/d and set N ˝ of positive N .x/ w.x/ N < 0. Take wO D w N on N measure we have the inequality j0 .v.x/I w.x// and wO D 0 on ˝ n N. We have wO 2 L1 .˝/d and Z Z j0 .v.x/I w.x// O .x/ w.x/ O d .x/ D j0 .v.x/I w.x// N .x/ w.x/ N d .x/ < 0; ˝

N

fwn g1 nD1 0

a contradiction with (3.39). Let be a countable dense set in Rd . By taking constant functions in place of w we get j .v.x/I wn / .x/ wn outside a certain S null set Mn . If follows that j0 .v.x/I wn / .x/ wn for all n outside a null sets 1 nD1 Mn . d 0 By Proposition 3.3 and density of fwn g1 in R it follows that .x/ z j .v.x/I z/ nD1 for all z 2 Rd on a set of full measure and the proof is complete. t u

3.8 Nemytskii Operator for Multifunctions In this section we prove a result, that will be useful in passing to the limit in the terms containing the Clarke subdifferential. Before we state the theorem, however, we need to recall a useful result on pointwise behavior of weakly convergence sequences (see Proposition 3.6 in [177]). To formulate it we will need the notion of Kuratowski–Painlevé upper limit of a sequence of sets An Rd , given by K- lim sup An D f x 2 Rd W x D lim xnk ; xnk 2 Ank ; 0 < n1 < n2 < : : : < nk < : : : g: n!1

k!1

Theorem 3.22. Let .; ˙; / be a -finite measure space, and let d be a positive natural number, and 1 p < 1. Let un ! u weakly in Lp ./d and un . / 2 G. / for -a.e. 2 and all n 2 N, where G. / Rd is a nonempty, compact set for -a.e. 2 . Then u.x/ 2 conv K- lim sup fun .x/g : n!1

Theorem 3.23. Let .; ˙; / be a -finite measure space and let W D Lp ./d with a positive natural number d and 1 < p < 1. Assume that the multifunction d h W Rd ! 2R satisfies the assumptions .h1/ for every s 2 Rd the multifunction ! h. ; s/ has a measurable selection, i.e., there exists a measurable function f W ˝ ! Rd such that f . / 2 h. ; s/ for -a.e. 2 , .h2/ the set h. ; s/ is nonempty, compact, and convex in Rd , for all s 2 Rd and for -a.e. 2 , .h3/ graph of s ! h. ; s/ is a closed set in R2d for -a.e. 2 , .h4/ we have jj c1 . / C c2 jsjp1 for all s 2 Rd , -a.e. 2 and all 2 h. ; s/, with c2 > 0 and c1 2 Lq ./, where 1q C 1p D 1.

3.8 Nemytskii Operator for Multifunctions

75 0

Then the multivalued operator HW W ! 2W defined by H.u/ D f 2 W 0 W . / 2 h. ; u. //

-a.e. on

g;

satisfies .H1/ .H2/ .H3/

H has nonempty and convex values, the graph of H is sequentially closed in .W; strong/ .W 0 ; weak/ topology, p1 kkW kc1 kLq ./ C c2 kukW for all w 2 W and 2 H.u/.

Proof. Let u 2 W. First we will prove .H3/. Let 2 H.u/. We have Z kk

W0

D sup kwkW D1

Z . / w. / d sup

kwkW D1

.c1 . / C c2 ju. /jp1 /jw. /j d :

Hence we have p1

p1

kkW 0 sup .kc1 kLq ./ kwkW C c2 kukW kwkW / D kc1 kLq ./ C c2 kukW ; kwkW D1

(3.40) and .H3/ is proved. We pass to the proof of .H1/. Let u 2 W. Consider a multifunction 3 ! h. ; u. // Rd . We will show that this multifunction has a measurable selection. Let vn W ! Rd be a sequence of simple (i.e., piecewise constant and measurable) functions such that jvn . /j ju. /j and vn . / ! u. / a.e. 2 . By .h1/ it follows that there exists the sequence of measurable functions n W ! Rd such that n . / 2 h. ; vn . // a.e. 2 . By (3.40) it follows that p1

kn kW 0 kc1 kLq ./ C c2 kvn kW

p1

kc1 kLq ./ C c2 kukW ;

and hence, for a subsequence, not renumbered, we have n !

weakly in W 0

for certain 2 W 0 :

(3.41)

Since for a.e. 2 we have jn . /j c1 . / C c2 jvn . /j c1 . / C c2 ju. /j; we can define G. / D fz 2 Rd W jzj c1 . / C c2 ju. /jg and we can use Theorem 3.22 concluding that a.e. 2 : . / 2 conv K- lim sup fn . /g n!1

Now z 2 K- lim supn!1 fn . /g, whenever nk . / ! z. Keeping in mind that vnk . / ! u. / and nk . / 2 h. ; vnk . // and we can use .h3/ to conclude that z 2 h. ; u. // for a.e. 2 . Hence we have . / 2 conv h. ; u. // D h. ; u. //

a.e.

2 ;

(3.42)

76


being the required measurable selection. Moreover, (3.41) implies that 2 W 0 and the proof of non-emptiness of H.u/ is complete. Convexity of H.u/ follows straightforwardly from the convexity of h. ; s/ in .h2/, and the proof of .H1/ is complete. Finally, we will establish .H2/. Let un ! u in W and n ! weakly in W 0 with n 2 H.un /. Passing to a subsequence, we may assume that un . / ! u. / for a.e.

2 and jun . /j f . / for a.e. 2 with f 2 Lp ./. Since we already know that 2 W we must prove that . / 2 h. ; u. // for a.e. 2 . The proof of this assertion exactly follows the lines of the proof of (3.42) in .H1/, which completes the proof of the theorem. t u Remark 3.4. The assertion .H2/ of the above Lemma also follows directly by the application of the convergence theorem of Aubin and Cellina (see Theorem 7.2.1 in [8] or Theorem 1.4.1 in [7]). We will use the notation q

H.u/ D Sh.;u.// ; as H.u/ is the set of all selections of the class Lq ./d of the multifunction 3 ! h. ; u. // Rd . The operator H is known as the Nemytskii operator for a multifunction h. Now we present a version of above theorem valid if the multifunction h has the form of a Clarke subdifferential. In such a case the above theorem simplifies. Note that for a functional of more than one variable, say '. ; s/, by @'. ; s/ we will always understand the Clarke subdifferential taken with respect to the last variable s. Theorem 3.24. Let .; ˙; / be a finite and complete measure space and denote W D Lp ./d with d 2 N and 1 < p < 1. Assume that the functional ' W Rd ! R satisfies the assumptions .'1/ for every s 2 Rd the function 3 ! '. ; s/ is measurable, .'2/ for -almost all 2 the function Rd 3 s ! '. ; s/ is locally Lipschitz, .'3/ jj c1 . / C c2 jsjp1 for all s 2 Rd , -a.e. 2 and all 2 @'. ; s/, with c2 > 0 and c1 2 Lq ./, where 1q C 1p D 1. 0

Then the multivalued operator HW W ! 2W defined by H.u/ D S@'.;u.// D f 2 W 0 W . / 2 @'. ; u. // q

-a.e. on

g

satisfies .H1/ H has nonempty and convex values, .H2/ the graph of H is sequentially closed in .W; strong/ .W 0 ; weak/ topology, p1 .H3/ we have kkW kc1 kLq ./ C c2 kukW for all w 2 W and 2 H.u/.

3.9 Clarke Subdifferential: Examples

77

Proof. We will use Theorem 3.23. The hypothesis .h4/ follows from .'3/ while the hypotheses .h2/ and .h3/ follow from the properties of the Clarke subdifferential (Propositions 3.4 and 3.5). It remains to prove .h1/. To this end consider the integral functional Z I.u/ D '. ; u. // d for u 2 Lp ./d :

We use Theorem 5.6.39 in [84] (see also Theorem 2.7.5 in [72], where, however, c1 is assumed to be a constant and not a function of ), whence it follows that I is locally Lipschitz and q

@I.u/ S@'.;u.// : Let s 2 Rd and let us . / D s for 2 . As the measure is finite, us 2 Lp ./d . Since, by Proposition 3.4, @I.us / is nonempty, then there exists 2 Lq ./d such that . / 2 @'. ; s/ for -a.e. 2 and the proof is complete. t u

3.9 Clarke Subdifferential: Examples As an example of the application of the Clarke subdifferential in PDEs let us consider the following example. Let ˝ be a bounded domain in Rd with Lipschitz 1 boundary and let g 2 Lloc .R/. Consider the following initial and boundary value problem @u.x; t/ u.x; t/ C g.u.x; t// D f .x; t/ @t

on

u.x; t/ D 0 on u.x; 0/ D u0 .x/

on

˝ .0; T/; @˝ .0; T/; ˝:

The question arises how to understand the weak solution of the above problem when g is discontinuous. It turns out that the function g can be replaced with its multivalued regularization constructed in the following way. First we define the functional Z s j.s/ D g.s/ ds for s 2 R: 0

As g is locally bounded, j must be locally Lipschitz and it makes sense to define @j.s/. We replace the above initial and boundary value problem by the following one

78


@u.x; t/ u.x; t/ C .x; t/ D f .x; t/ @t

on

˝ .0; T/;

.x; t/ 2 @j.u.x; t//

on

˝ .0; T/;

u.x; t/ D 0 on u.x; 0/ D u0 .x/

on

@˝ .0; T/; ˝:

This problem is called a multivalued partial differential equation, a partial differential inclusion, or a subdifferential inclusion and it turns out, that under appropriate assumptions of f ; u0 and the growth condition on @j it has a weak solution understood in the following sense. Find u 2 L2 .0; TI H01 .˝// with u0 2 L2 .0; TI H 1 .˝// and 2 L2 .0; TI L2 .˝// 2 such that u.0/ D u0 , .t/ 2 [email protected]// , and for a.e. t 2 .0; T/ and all v 2 H01 .˝/, hu0 .t/; viH 1 .˝/H 1 .˝/ C .ru.t/; rv/L2 .˝/d C ..t/ f .t/; v/L2 .˝/ D 0; 0

2 we mean the set of all functions h 2 L2 .˝/ such Recall that for v 2 L2 .˝/ by [email protected]/ that h.x/ 2 @j.v.x// a.e. in ˝. If we identify L2 .0; TI L2 .˝// D L2 .˝ .0; T//, 2 2 we can also write [email protected]// D [email protected].;t// . We can associate with the above problem the following problem known as hemivariational inequality. Find u 2 L2 .0; TI H01 .˝// with u0 2 L2 .0; TI H 1 .˝// such that u.0/ D u0 , and for a.e. t 2 .0; T/ and all v 2 H01 .˝/,

hu0 .t/; viH 1 .˝/H 1 .˝/ C .ru.t/; rv/L2 .˝/d 0 Z C j0 .u.x; t/I v.x// dx .f .t/; v/L2 .˝/ : ˝

It follows directly from the definition (3.37) that every solution of above partial differential inclusion is also a solution of the hemivariational inequality. The converse also holds true, but this has been shown only recently by Carl (see [49]): his proof is more involved and uses the notion of upper and lower solutions. The Clarke subdifferential is very easy to compute in finite dimensions using the characterization (3.38), when the number of points in which the functional is not continuously differentiable is a null Lebesgue set. Example 1. Let us consider the example of the functional ' W R ! R 8 ˆ ˆ
for s < 1; for s 2 Œ1; 1; for s > 1:

3.9 Clarke Subdifferential: Examples

79

This functional is of class C1 on a neighborhood of every s 62 f1; 1g, and for the points f1; 1g we take convex hull of left and right derivatives 8 ˆ f1g ˆ ˆ ˆ ˆ ˆ ˆŒ1; 1 < @'.s/ D f1g ˆ ˆ ˆ ˆ Œ1; 1 ˆ ˆ ˆ :f1g

for

s < 1;

for

s D 1;

for

s 2 .1; 1/;

for

s D 1;

for

s > 1:

The potential and its subdifferential are both depicted in Fig. 3.1.

1,5

1,5

ϕ(s)

1 0,5 −2

−1,5

−1

−0,5

0

∂ϕ(s)

1 0,5 0

0,5

1

1,5

S 2 −2

−1,5

−1

−0,5

−0,5

0

0

0,5

1

1,5

S2

−0,5

−1

−1

−1,5

−1,5

Fig. 3.1 The example of a simple locally Lipschitz function, which is nondifferentiable in two points, and its Clarke subdifferential

Example 2. Let W Œ0; 1/ ! R be a continuous function such that .0/ > 0. Let s0 2 Rd . Define the functional ' W Rd ! R by the formula Z '.s/ D

jss0 j 0

.r/ dr:

(3.43)

It is easy to verify, using the mean value theorem for integrals, that ' is locally Lipschitz. It is also easy to calculate the gradient of ' for all s ¤ s0 , which establishes the fact that ' is of class C1 on the neighborhood of every s ¤ s0 . The value of the Clarke subdifferential at s0 is obtained by taking the convex hull of the sphere obtained from limits of the gradients of all sequences converging to s0 . We get 8n o < .js s j/ ss0 0 jss0 j @'.s/ D :fy 2 Rd W jyj .0/g D B.0; .0//

for s ¤ s0 ; for s D s0 :

Graphs of ' and @' for d D 1, s0 D 1, and .r/ D 1 r are depicted in Fig. 3.2.

80

3 Mathematical Preliminaries 0,6

1,5

ϕ(s)

0,5

1

0,4

0,5

0,3 −1

0,2

−0,5

−0,5

0

S 0

0,5

1

1,5

2

2,5

3

−0,5

0,1 −1

0

∂ϕ(s)

−1 0

0,5

1

1,5

2

2,5

−1,5

S3

Fig. 3.2 The second example of a nondifferentiable nonconvex potential and its Clarke subdifferential

Example 3. Note that it is not required in Example 2 for to be a continuous function. For example, we can consider the following function ( .r/ D

1

for

r 2 Œ0; M/;

2

for

r 2 ŒM; 1/;

where 1 ; M > 0, and 2 2 R. Then, ' calculated by (3.43) is given by ( '.s/ D

1 js s0 j

for js s0 j 2 Œ0; M/;

1 M C 2 .js s0 j M/

for js s0 j 2 ŒM; 1/:

For s ¤ s0 and js s0 j ¤ M the functional ' is continuously differentiable. Using this fact and the characterization (3.38) we can calculate its Clarke subdifferential as 8 ˆ ˆB.0; ˆ n 1 / o ˆ ˆ < 1 ss0 jss0 j @'.s/ D ˚ ˆ .s s0o/ W 2 Œ 1 ; 2 ˆ M ˆ n ˆ ˆ : 2 ss0 jss0 j

for s D s0 ; for js s0 j 2 .0; M/; for js s0 j D M; for js s0 j 2 .M; 1/:

The plot of ' and its Clarke subdifferential for d D 1, s0 D 0, 1 D 2, 2 D 1, and M D 1 is presented in Fig. 3.3. 3,5

j(s)

3 2,5 2 1,5

−2

1 0,5 −2

−1,5

−1

−0,5

0

S 0

0,5

1

1,5

2

−1,5

−1

2,5 ∂j(s) 2 1,5 1 0,5 0 −0,5 0 0,5 −0,5 −1 −1,5 2 −2,5

S 1

1,5

2

Fig. 3.3 The third example of a locally Lipschitz potential and its Clarke subdifferential


81

Finally, we present a brief comparison of the Clarke subdifferential with the convex subdifferential. If the functional ' W X ! R [ f1g, where X is a Banach space, is convex, then its convex subdifferential is defined as ( @conv '.s/ D

;

if s … dom '; 0

f 2 X W h; v siX 0 X '.v/ '.s/ 8 v 2 dom 'g otherwise;

where dom ' is the set of all points in which ' in not equal to 1. If a functional ' W X ! R is both convex and locally Lipschitz then both subdifferentials must coincide (see, e.g., Proposition 2.2.7 in [72]). There are, however, examples where only one of the two notions makes sense. For the functionals which are locally Lipschitz and nonconvex we can speak only of the Clarke subdifferential, while for convex functionals which assume infinite values (and such functionals are useful, for example, in constrained optimization or in unilateral problems in solid mechanics) only the convex subdifferential makes sense.

3.10 Comments and Bibliographical Notes The preliminary material provided in this chapter (and much more) can be found, for example, in the following monographs devoted to the Navier–Stokes equations: [13, 75, 88, 99, 108, 146, 156, 157, 218, 219, 221]. More information about differential inclusions and theory of multifunctions can be found in [7, 8]. For the theory of Clarke subdifferential and hemivariational inequalities we refer the reader to [50, 71, 72, 114, 173, 177, 184]. The applications of this theory in contact mechanics are described in [177, 217].

4

Stationary Solutions of the Navier–Stokes Equations

Now I think hydrodynamics is to be the root of all physical science, and is at present second to none in the beauty of its mathematics. – William Thomson, 1st Baron Kelvin

In this chapter we introduce some basic notions from the theory of the Navier– Stokes equations: the function spaces H, V, and V 0 , the Stokes operator A with its domain D.A/ in H, and the bilinear form B. We apply the Galerkin method and fixed point theorems to prove the existence of solutions of the nonlinear stationary problem, and we consider problems of uniqueness and regularity of solutions.

4.1 Basic Stationary Problem Let Q D Œ0; L3 and define the spaces H D fu D .u1 ; u2 ; u3 / 2 LP p2 .Q/3 W div u D 0g; and P p1 .Q/3 W div u D 0g; V D fu D .u1 ; u2 ; u3 / 2 H with the norms sZ ju.x/j2 dx;

kukH D juj D

juj2 D .u; u/;

Q

and sZ jru.x/j2 dx;

kukV D kuk D

kuk2 D .ru; ru/;

Q


83

84

4 Stationary Solutions of the Navier–Stokes Equations

respectively. V and H are Hilbert spaces and V H. If we identify H with its dual H 0 in view of the Riesz–Fréchet representation theorem, we have V H V 0 with continuous embeddings and with each space dense in the following one.

4.1.1 The Stokes Operator We define the Stokes operator A W V ! V 0 by Z hAu; vi D ru rv dx for all

v 2 V:

(4.1)

Q

Using this definition we can write the weak formulation of the Stokes problem: Given f 2 V 0 , find u 2 V and p 2 LP 2 .Q/ such that u C rp D f ;

div u D 0;

(4.2)

in the weak sense, that is, .ru; rv/ .p; div v/ D hf ; vi

for all

P p1 .Q/3 ; v2H

(4.3)

for all

v 2 V:

(4.4)

in an equivalent abstract form hAu; vi D hf ; vi

We shall prove that for f 2 H, the Stokes operator coincides with the minus Laplacian, Au D u. To this end we use the explicit spectral representation of functions in LP 2 .Q/3 . Assume in the beginning that f 2 LP 2 .Q/3 . Then f .x/ D

X

X

x

fk e2ik L ;

k2Z 3

fk D fNk ;

jfk j2 < 1;

f0 D 0;

(4.5)

k2Z 3

where fk D .fk1 ; fk2 ; fk3 /. Let u 2 V and p 2 LP 2 .Q/ satisfy (4.3). Assuming that u.x/ D

X

x

uk e2ik L ;

p.x/ D

k2Z 3

X

x

pk e2ik L

k2Z 3

with u0 D 0, k uk D 0, p0 D 0, uk D uN k , pk D pN k we obtain,

2ik 4 2 jkj2 uk C pk D fk : 2 L L

Multiplying this equation by k and using the relation k uk D 0 we obtain pk D

L fk k 2i jkj2

for

k 6D 0;

(4.6)

4.1 Basic Stationary Problem

85

and then we compute 1 L2 uk D 4 2 jkj2

fk k fk k 2 jkj

for

k 6D 0:

Now, if f 2 H, then fk k D 0 and the equation for uk reduces to uk D

1 L2 fk 4 2 jkj2

for

k 6D 0:

Thus, the weak solution u of the Stokes problem (4.4) (with D 1) coincides with the weak solution of the problem u D f for f 2 H. From this we see P p2 .Q/3 . Moreover, from the above explicit formulas for the immediately that u 2 H Fourier coefficients uk and pk we have the following regularity theorem. P p2 .Q/3 , p 2 H P p1 .Q/, and Theorem 4.1. If f 2 LP p2 .Q/3 , then u 2 H kuk2HP 2 .Q/3 C kpk2HP 1 .Q/ 0

0

L2 4 2

L2 C 1 kf k2LP 2 .Q/3 : 22

Proof. We have kuk2HP 2 .Q/3 D 0

X

jkj4 juk j2 D

k2Z 3

2

jkj4

k2Z 3

X

jkj4

k2Z 3

D

X

ˇ ˇ2 ˇ ˇ 1 L4 ˇfk k fk k ˇ ˇ 2 16 4 jkj4 jkj2 ˇ

X 1 L4 L4 2 4 1 jf j C 2 jkj jfk j2 k 2 16 4 jkj4 2 16 4 jkj4 3 k2Z

4

L kf k2LP 2 .Q/3 ; 4 4 2

and kpk2HP 1 .Q/ D p

X

jkj2 jpk j2 D

k2Z 3

X k2Z 3

jkj2

L2 4 2

ˇ ˇ ˇ fk k ˇ2 ˇ ˇ ˇ jkj2 ˇ

X L2 L2 jfk j2 D kf k2LP 2 .Q/3 ; 2 2 4 4 3

k2Z

which ends the proof.

t u

Remark 4.1. Observe that setting f , u, and p of the form (4.5) and (4.6), respectively, directly to Eqs. (4.2) we would get the same spectral representations of the solution u, p. Thus, in particular, P p2 .Q/3 : D.A/ D fu 2 V W Au 2 Hg D V \ H

86


4.1.2 The Nonlinear Problem Let us consider the stationary problem: u C .u r/u C rp D f

in Q ;

div u D 0

in Q ;

with one of the boundary conditions: 1. Q D Œ0; L3 in R3 and we assume periodic boundary conditions, or 2. Q is a bounded domain in R3 , with smooth boundary, and we assume the homogeneous Dirichlet boundary condition u D 0 on @Q. In the second case we define H D fu D .u1 ; u2 ; u3 / 2 L2 .Q/3 W div u D 0; un D u nj@Q D 0g; and V D fu D .u1 ; u2 ; u3 / 2 H01 .Q/3 W div u D 0g ; with the norms sZ ju.x/j2 dx;

kukH D juj D

juj2 D .u; u/;

Q

and sZ jru.x/j2 dx;

kukV D kuk D

kuk2 D .ru; ru/:

Q

The Stokes operator A is defined in the same way as in the periodic case (cf. (4.1)) and we have D.A/ D fu 2 V W Au 2 Hg D V \ H 2 .Q/3 : For both cases, let us denote b.u; v; w/ D ..u rv/; w/ for u; v; w 2 V and define a nonlinear operator B W V ! V 0 by .Bu; v/ D b.u; u; v/ for all v 2 V. We shall also write ..u; v// instead of .ru; rv/. Exercise 4.1. Prove that for all u; v; w 2 V it is b.u; v; w/ D b.u; w; v/; and thus b.u; v; v/ D 0.


87

Exercise 4.2. Prove that for all u; v; w 2 V, jb.u; v; w/j C.Q/kukkvkkwk: Hint. Use the Hölder inequality and the continuous embedding of L4 .Q/ in V. Let H and V be the corresponding function spaces. Then the weak formulation of the above nonlinear stationary problem is as follows. Problem 4.1. For f 2 H (or f 2 V 0 ) find u 2 V such that ..u; v// C b.u; u; v/ D hf ; vi

for all

v2V

or, equivalently, Au C Bu D f

in H

or

V 0:

Remark 4.2. Concerning regularity of solutions one can prove that if f 2 H then all solutions u belong to D.A/. We shall prove the following existence theorem. Theorem 4.2. Let Q D Œ0; L3 (for the periodic problem) or let Q be a bounded domain in R3 with smooth boundary (for the Dirichlet problem). Then (i) For every f 2 V 0 and > 0 there exists at least one solution of Problem 4.1. (ii) If 2 c1 .Q/kf kV 0 , then the solution of Problem 4.1 is unique. Proof. ad .i/. To prove existence of solutions we use the Galerkin method. For every m 2 N let um .x/ D

m X

j;m !j .x/;

j;m 2 R;

jD1

be the approximate solution, where !j , j 2 N are the eigenfunctions of the Stokes operator corresponding to the considered problem. This means that ..um ; v// C b.um ; um ; v/ D hf ; vi

for all

v 2 Vm ;

(4.7)

where Vm D spanf!1 ; : : : ; !m g. Such solution exists in view of the Brouwer fixed point theorem. Exercise 4.3. Prove that the operator A W H D.A/ ! H is a self-adjoint positive operator in H and is an isomorphism from D.A/ onto H, and its inverse A1 W H ! H is continuous, compact, and self-adjoint so there exists a sequence j , j 2 N, 0 < 1 2 3 : : :, j ! 1, and a family of elements !j 2 D.A/ orthonormal in H such that A!j D j !j for j 2 N, cf. Theorem 3.15. Exercise 4.4. Prove existence of um using the Brouwer fixed point theorem.

88


Solution to Exercise 4.4. Consider the ball B D fu 2 Vm W kuk 1 kf kV 0 g in Vm and the map ˚ W uO ! u where uO 2 B and u is the unique solution of the linear problem ..u; v// C b.Ou; u; v/ D hf ; vi

for all v 2 Vm :

Setting v D u in this identity we obtain that u 2 B. We shall show that ˚ is continuous in Vm . To this end, let ..w; v// C b.w; O u; v/ D hf ; vi

for all

v 2 Vm :

Subtracting the equation for w from that for u, setting v D u w, and using the estimate for the solution v, we obtain ku wk

C.Q/ kf kV 0 kOu wk O ; 2

which proves the continuity of ˚. As Vm is a finite dimensional space and ˚ is a continuous map from a convex and compact set B to B, we conclude existence of um , solution of (4.7) in view of the Brouwer fixed point theorem. From (4.7) we conclude kum k

kf kV 0 :

Thus, for a subsequence, um ! u

weakly in V;

and um ! u strongly in H; as V is compactly embedded in H. Passing with m to the limit in Eq. (4.7) we obtain equation ..u; v// C b.u; u; v/ D hf ; vi for all v being a finite linear combination of elements of the basis !k , k 2 N. Since such elements v are dense in V, the existence part of the theorem has been proved. Proof of .ii/. To prove uniqueness of solutions for large viscosity coefficients with respect to mass forces, 2 > c1 .Q/kf kV 0 , let us assume that there are two distinct solutions u1 and u2 , that is, ..u1 ; v// C b.u1 ; u1 ; v/ D hf ; vi; and ..u2 ; v// C b.u2 ; u2 ; v/ D hf ; vi


89

for all v 2 V. Setting v D u1 u2 and subtracting the second equation from the first one we obtain ku1 u2 k2 D b.u1 u2 ; u2 ; u1 u2 / c1 .Q/ku1 u2 k2 ku2 k

c1 .Q/ kf kV 0 ku1 u2 k2 ;

as ku2 k 1 kf kV 0 , whence c1 .Q/ 0 kf kV ku1 u2 k2 0 ; which gives a contradiction. Thus u1 D u2 .

t u

Exercise 4.5. Prove that Theorem 4.2 is valid also in the two-dimensional case, when Q R2 is a square (in the periodic case) or a bounded domain with smooth boundary (in the case of the Dirichlet boundary conditions). In the end we shall show how one can proceed differently to prove existence of the Galerkin approximate solutions um satisfying (4.7). Theorem 4.3. Let X be a finite dimensional Hilbert space, with a scalar product Œ; and the associated norm Œ, and let P W X ! X be a continuous map such that ŒP./; > 0 for

Œ D k > 0

for some k. Then there exists 2 X with Œ k for which P./ D 0. Proof. Let B D B.0; k/ be a closed ball in X, centered at zero and with radius k. Assume that P is different from zero in this ball. Then the map ! S./ D

kP./ ; ŒP./

SWB!B

(4.8)

is continuous in X. From the Brouwer fixed point theorem it follows that S has a fixed point in B, that is, 0 D

kP.0 / ŒP.0 /

(4.9)

for some 0 2 B. We have Œ0 D k. Multiplying (4.9) by 0 we get Œ0 2 D

kŒP.0 /; 0 ; ŒP.0 /

which contradicts ŒP./; > 0 for Œ D k. Thus P./ D 0 for some 2 B.

t u

90


Now, let X D Vm with the norm induced from V. Let us define P D Pm W Vm ! Vm by the relation, ŒPm .u/; v D ..Pm .u/; v// D ..u; v// C b.u; u; v/ hf ; vi

(4.10)

for all u; v 2 Vm . The map is well defined in view of the Riesz–Fréchet representation theorem. In fact, for every u in Vm the map v ! ..u; v// C b.u; u; v/ hf ; vi defines a linear and continuous functional on Vm . Thus, there exists a unique Pm .u/ in Vm satisfying the above relation. The map Pm is continuous in Vm and ŒPm .u/; u D kuk2 C b.u; u; u/ hf ; ui D kuk2 hf ; ui kuk2 kf kV 0 kuk D kukfkuk kf kV 0 g: kf k

Whence, for k > V 0 and kuk D k we have ŒPm .u/; u > 0. In view of our auxiliary theorem, there exists um in Vm such that Pm .um / D 0, that is, (4.7) is satisfied. t u Exercise 4.6. Prove that the maps S defined in (4.8) and Pm defined in (4.10) are continuous.

4.1.3 Other Topological Methods to Deal with the Nonlinearity In this subsection we shall use the Schauder and, alternatively, the Leray–Schauder fixed points theorems to prove directly the existence of a solution of the nonlinear problem, thus bypassing the Galerkin approximate solutions. When the viscosity is large enough with respect to external forces, we shall use the Banach contraction principle to prove the existence of the unique solution of the nonlinear problem. Exercise 4.7. Let f 2 V 0 , Q R3 as above. Prove that there exists u 2 V such that ..u; v// C b.u; u; v/ D hf ; vi

for all

v2V

by using (a) the Schauder fixed point theorem, (b) the Leray–Schauder fixed point theorem, (c) assuming that the viscosity is large enough with respect to external forces, use the Banach contraction principle to prove the unique solution u. Solution to Exercise 4.7. (a) Let us define a map T in V by ..Tu; v// C b.u; Tu; v/ D hf ; vi

for all

v 2V:


91

Using the Lax–Milgram lemma we conclude that the map T is well defined. Setting v D Tu we get kTuk2 D hf ; Tui kf kV 0 kTuk ; whence kTuk

1 kf kV 0 r:

Define K D fv 2 V W kvk rg. Then T.K/ K. Let us consider the map T W K ! K. We shall show that T is continuous and compact in the topology of V. Let u1 , u2 be in K. We have, ..Tu1 ; v// C b.u1 ; Tu1 ; v/ D hf ; vi for all

v 2 V;

..Tu2 ; v// C b.u2 ; Tu2 ; v/ D hf ; vi for all

v 2 V:

and

Subtracting the second equation from the first one we get ..Tu1 Tu2 ; v// C b.u1 u2 ; Tu2 ; v/ C b.u1 ; Tu1 Tu2 ; v/ D 0 for all v in V. Set v D Tu1 Tu2 and we get, since b.u1 ; Tu1 Tu2 ; Tu1 Tu2 / D 0, kTu1 Tu2 k

C C1 kTu2 kju1 u2 jL4 .Q/3 rku1 u2 k:

The continuity follows immediately. To prove compactness observe that any sequence .uk / K contains a subsequence which converges strongly in the L4 topology. Thus, compactness follows easily from the first of the above inequalities. (b) Let us define the map u ! Tu in V by ..Tu; v// C b.u; u; v/ D hf ; vi

for all

v 2V:

(4.11)

Using the Riesz–Fréchet representation theorem we conclude that the map T is well defined. Assume that Tu D u, where 2 Œ0; 1. We shall prove that all such u are in some ball. If D 0, then u D 0. Now, for > 0, Tu D 1 u. Setting v D u in (4.11) we obtain kuk2 kf kV 0 kuk, whence kuk 1 kf kV 0 M. Now, we shall prove continuity and compactness of T in V. Let um ; uk be in V. Then, subtracting the equations for Tum and Tuk , and setting v D Tum Tuk we obtain

92


kTum Tuk k2 D b.um ; Tum Tuk ; um uk / C b.um uk ; Tum Tuk ; uk / jum jL4 .Q/3 kTum Tuk kjum uk jL4 .Q/3 C jum uk jL4 .Q/3 kTum Tuk kjuk jL4 .Q/3 ; whence, 1 .jum jL4 .Q/3 C juk jL4 .Q/3 /jum uk jL4 .Q/3 C .jum jL4 .Q/3 C juk jL4 .Q/3 /kum uk k:

kTum Tuk k

To prove the continuity of T in V observe that if um ! uk in V then from the above estimate, kTum Tuk k ! 0. To prove the compactness of T in V, let fum g be a bounded sequence in V. Then, it contains a convergent subsequence fu g in L4 .Q/3 . Thus, from the above estimate it follows that the sequence fTu g is convergent. Using the Leray–Schauder fixed point theorem we conclude that there exists u in V such that Tu D u. This u is of course a solution of our stationary Navier–Stokes problem. (c) Let us define the map T W V ! V by ..Tu; v// C b.u; Tu; v/ D hf ; vi for all

v 2 V:

(4.12)

The map is well defined. We shall prove that it is contracting in a ball B.0; r/ in V provided the viscosity is large enough with respect to external forces. Let ..Tu1 ; v// C b.u1 ; Tu1 ; v/ D hf ; vi for all

v 2 V:

We get ..Tu Tu1 ; v// C b.u; Tu Tu1 ; v/ C b.u u1 ; Tu1 ; v/ D 0 for all v 2 V, whence, with v D Tu Tu1 , kTu Tu1 k

C kTu1 kku u1 k:

From (4.12) it follows that for every u 2 V, kTuk 1 kf kV 0 . Therefore, kTu Tu1 k

C kf kV 0 ku u1 k: 2

Consider the closed ball B D B.0; r/ in V with r D 1 kf kV 0 . Assume moreover that C2 kf kV 0 < 1. Then the map T has the following properties: T W B ! B and is contracting on B. By the Banach contraction principle, there exists a unique u 2 B.0; r/ such that Tu D u. This u is the unique solution of the stationary Navier– Stokes problem.


93

Exercise 4.8. Prove the Ladyzhenskaya inequality (3.8) for all functions u 2 P p1 .Q/, Q D Œ0; L2 , knowing that it holds for all functions u 2 H 1 .˝/, where H 0 ˝ is a bounded set in R2 . Exercise 4.9. Using the methods from Exercise 4.7 prove existence of solutions of the stationary Navier–Stokes problems in the two-dimensional case, for periodic and homogeneous boundary conditions, respectively. You may find useful the Ladyzhenskaya inequality (3.8).

4.2 Comments and Bibliographical Notes There are many other important and interesting problems concerning stationary solutions of the Navier–Stokes equations. We mention problems of flows with nonhomogeneous Dirichlet boundary conditions or with other types of boundary conditions, flows in nonsmooth, unbounded, or exterior domains. Still other problems concern behavior of solutions when the viscosity tends to zero, bifurcation of solutions, and the structure of the set of stationary solutions. In Chaps. 5 and 6 we shall consider stationary flows appearing in applications in the theory of lubrication. For thorough treatment of the basic mathematical problems of the Navier–Stokes equations we refer the reader to monographs: [75, 88, 99, 108, 146, 156, 218, 219], and, for the compressible case, to [157, 187].

5

Stationary Solutions of the Navier–Stokes Equations with Friction

In this chapter we consider the three-dimensional stationary Navier–Stokes equations with multivalued friction law boundary conditions on a part of the domain boundary. We formulate two existence theorems for the formulated problem. The first one uses the Kakutani–Fan–Glicksberg fixed point theorem, and the second one, with the relaxed assumptions, is based on the cut-off argument.

5.1 Problem Formulation Let ˝ R3 be a bounded domain with Lipschitz boundary as schematically presented in Fig. 5.1. The boundary @˝ is divided into two disjoint parts @˝ D D [ C . Assume that the viscosity is a positive constant and the mass force density f does not depend on time. Consider the stationary system of the Navier– Stokes equations u C .u r/u C rp D f in ˝;

(5.1)

div u D 0 in ˝;

(5.2)

with the following boundary conditions. On D we assume the Dirichlet boundary condition u D 0. Moreover we let un D 0 on C ; T 2 h.u / on C ;

(5.3) (5.4)

where un D u n is the outward normal component of the velocity vector, its tangent component is given by u D u un n on C , and T D Tn .Tn n/n is the tangential part of the stress vector Tn © Springer International Publishing Switzerland 2016 G. Łukaszewicz, P. Kalita, Navier–Stokes Equations, Advances in Mechanics and Mathematics 34, DOI 10.1007/978-3-319-27760-8_5

95

96

5 Stationary Solutions of the Navier–Stokes Equations with Friction

T D pI C 2D.u/;

(5.5)

3

[cf. (2.24)–(2.25)], and h W C R3 ! 2R is a friction multifunction.

Fig. 5.1 The domain of the studied problem. The set ˝ is a space between two cylinders. The Dirichlet boundary is the wall of the outer cylinder, and two rings bounding ˝ at top and bottom. The contact boundary C is the wall of the inner, rotating, cylinder. On D we impose the homogeneous Dirichlet boundary condition, and on C we impose the frictional contact condition which allows for the occurrence of either stick, or slip between the liquid and the rotating cylinder

Exercise 5.1. Prove that T D 2D.u/ on C , where D.u/ D D.u/n D.u/n n.

5.2 Friction Operator and Its Properties We will prove the existence of a weak solution for two cases. In the first case, in Sect. 5.4 we will assume the linear growth condition on the friction multifunction h (see .h4/ below) with the restriction on the growth constant (see .h5/ below) and we will use the Kakutani–Fan–Glicksberg theorem to obtain the solution. Next, in Sect. 5.5 we will relax the conditions on h, namely instead of the linear growth condition we will assume the weaker growth condition with exponent p 1 where 2 p < 4 (see .h6/ below) This possibility comes from the fact that in three space dimensions the trace operator is compact from the space H 1 .˝/ to Lp .@˝/ with 1 p < 4. Moreover we will need no restriction on the constant in the growth condition. This restriction is clearly nonphysical, i.e., it gives a limitation on the magnitude of the boundary friction force no matter if this force is “dissipative” (i.e., directed opposite to the velocity) or “excitatory” (i.e., directed the same as the velocity). We will replace .h5/ with its one sided counterpart denoted by .h7/ below. This condition enforces the limitation on the friction force only if it has the same direction as the velocity. The proof of existence for the relaxed assumptions will use the cut-off argument.

5.2 Friction Operator and Its Properties

97

We assume the following properties of the friction multifunction h: .h1/ for every s 2 R3 the multifunction x ! h.x; s/ has a measurable selection, i.e., there exists gs W C ! R3 , a measurable function, such that gs .x/ 2 h.x; s/ for all x 2 C , .h2/ the set h.x; s/ is nonempty, compact, and convex in R3 , for all s 2 R3 and for a.e. x 2 C , .h3/ the graph of s ! h.x; s/ is closed in R6 for a.e. x 2 C , .h4/ for all s 2 R3 , a.e. x 2 C and all 2 h.x; s/, we have jj c1 .x/ C c2 jsj, with some c2 0 and c1 2 L2 .C /. Denote W D L2 .C /3 and define a multivalued operator HW W ! 2W by H.u/ D f 2 W W .x/ 2 h.x; u.x//

a.e. on

C g:

(5.6)

The following Lemma is a straightforward consequence of Theorem 3.23. Lemma 5.1. Assume the multifunction h satisfies .h1/–.h4/. Then the operator H given by (5.6) satisfies .H1/ .H2/ .H3/

H has nonempty and convex values, the graph of H is sequentially closed in .W; strong/ .W; weak/ topology, for all w 2 W and 2 H.u/, kkW kc1 kL2 .C / C c2 kukW .

For the first existence result we will need the additional bound on the constant c2 in the growth condition .h4/, dependent on the viscosity coefficient and the norm of the trace operator k k, .h5/ c2 k k2 < 2. For the second existence result, the assumptions .h4/ and .h5/ will be replaced by the following, weaker ones: .h6/ there exists 2 p < 4 such that for all s 2 R3 we have max jj c3 .x/ C c4 jsjp1 for a.e. x 2 C ;

2h.x;s/

with some c3 2 L2 .C / and c4 0, and h 2 and d2 2 L2 .C / we have .h7/ For all s 2 Rd , some d1 2 0; kk 2 min s d1 jsj2 d2 .x/ for a.e. x 2 C :

2h.x;s/

When .h4/ is replaced by .h6/, the multivalued friction operator is defined on Lp .C /3 in place of L2 .C /3 . We will denote U D Lp .C /3 . Then U 0 D Lq .C /3 0 with 1p C 1q D 1 and q 2 43 ; 2 . Define the multifunction G W U ! 2U by G.u/ D f 2 U 0 W .x/ 2 h.x; u.x// for a.e. x 2 C g:

(5.7)

Analogously to Lemma 5.1 we have the following result which is a straightforward consequence of Theorem 3.23.

98


Lemma 5.2. Assume the multifunction h satisfies .h1/–.h4/. Then the operator G given by (5.7) satisfies .G1/ .G2/ .G3/

G has nonempty and convex values, the graph of G is sequentially closed in .U; strong/ .U 0 ; weak/ topology, for all w 2 W and 2 G.u/, kkU0 kc3 kLq .C / C c4 kukU .

5.3 Weak Formulation Let VQ D fu 2 C1 .˝/3 W div u D 0; u D 0 on D ; un D 0 on C g and V D closure of VQ in H 1 .˝/3 norm; W D L2 .C /3 : If v 2 V then we have the following Korn inequality (see, for instance, [194] for a proof of this inequality in a general setting) kvk2H 1 .˝/3 cK

Z

jD.u/j2 dx:

(5.8)

˝

Note that the Korn inequality implies that k k, where sZ jD.u/j2 dx

kuk D

(5.9)

˝

is a norm on the space V equivalent to the standard H 1 .˝/3 norm. We denote by W V ! W the trace operator and by k k its norm k kL .VIW/ . We can pass to the derivation of the weak formulation of the studied problem. To this end we assume that u, p, and T are smooth enough and satisfy (5.1)–(5.4). We multiply (5.1) by v 2 VQ and integrate over ˝, which gives Z

Z

˝

ui;jj vi dx C

˝

Z uj ui;j vi dx C

Z ˝

p;i vi dx D

˝

fi vi dx:

Integrating by parts the term with pressure we obtain Z

Z ˝

p;i vi dx D

Z ˝

pvi;i dx C

@˝

pvn dS:

(5.10)

5.3 Weak Formulation

99

Using the fact that div v D 0 and vn D 0 on the whole @˝, it follows that Z ˝

p;i vi dx D 0:

(5.11)

As div u D 0, we have, for smooth u, that uj;ij D 0 and hence ui;jj vi D .ui;jj C uj;ij /vi D .ui;j C uj;i /;j vi : From the following Green formula valid for a smooth matrix valued function F (F W ˝ ! M 33 ) and v 2 V Z Z Fij;j vi C Fij vi;j dx D Fij vi nj dS; (5.12) ˝

@˝

we have Z

Z

˝

ui;jj vi dx D 2 Z D 2

D.u/ij;j vi dx

˝

˝

Z

D.u/ij vi;j dx 2

Z D 2

˝

Z D.u/ij vi;j dx

C

@˝

D.u/n vn C D.u/ v dS

T v dS:

Finally, by the symmetry of D.u/ Z ˝

D.u/ij vi;j dx D

1 2

Z ˝

D.u/ij vi;j dx C

1 2

Z ˝

D.u/ji vj;i dx

Z Z 1 1 D D.u/ij vi;j dx C D.u/ij vj;i dx 2 ˝ 2 ˝ Z D D.u/ij D.v/ij dx:

(5.13)

˝

In this way we have proved that Z

˝

Z ui;jj vi dx D 2

˝

Z D.u/ij D.v/ij dx

C

T v dS:

(5.14)

Exercise 5.2. Prove the Green formula (5.12). Using (5.11) and (5.14) in (5.10) we can justify the following weak formulation of the considered problem.

100


Problem 5.1. Let f 2 L2 .˝/3 . Find u 2 V and 2 H.u / such that for every v 2 V Z

Z 2

˝

D.u/ij D.v/ij dx C

Z

C

v dS C

Z

˝

.u r/u v dx D

Remark 5.1. If C is a piecewise flat surface (i.e., C D on each Ck ), then for u; v 2 V Z 2

SK kD1

f v dx:

˝

(5.15)

Ck , where n D const

Z D.u/ij D.v/ij dx D

˝

ru rv dx;

(5.16)

˝

and (5.15) could be replaced by Z

Z

Z

ru rv dx C ˝

C

v d C

Z

˝

.u r/u v dx D

˝

f v dx:

Q On each Ck Let us prove (5.16). To this end first assume that u 2 V and v 2 V. we can construct an orthonormal basis n; 1 ; 2 , where both vectors 1 ; 2 lie in the Q On Ck we have rvn D @vn nC @vn 1 C @vn 1 D @vn n, where surface Ck . Take v 2 V. @n @1 @1 @n the last equality follows from the fact that vn D 0 on Ck . Now, we have Z

Z ˝

uj;i vi;j dx D

Z ˝

uj vi;ij dx C

@˝

uj vi;j ni dS:

As div v D 0, and u D 0 on D , we have Z

Z

Z

@vn n dS D uj;i vi;j dx D uj vi;j ni dS D u @n ˝ C C

Z C

.un n C u /

@vn n dS D 0: @n

The last equality follows from the fact that un D 0 and u is orthogonal to n on C . Using (5.13) we have Z

Z ˝

D.u/ij D.v/ij dx D

˝

1 D 2

D.u/ij vi;j dx D

Z ˝

1 2

Z

Z ˝

ui;j vi;j dx C

˝

uj;i vi;j dx

ui;j vi;j dx;

and the assertion follows by the density of VQ in V. We introduce the linear operator A W V ! V 0 and the bilinear operator B W V V ! V 0 given by Z hAu; vi D 2

˝

D.u/ij D.v/ij dx

for

u; v 2 V;

(5.17)

5.3 Weak Formulation

101

and Z hB.u; w/; vi D

˝

.u r/w v dx

for

u; w; v 2 V;

respectively, and the functional F 2 V 0 given by Z hF; vi D

˝

f v dx

for

v 2 V:

Note that, for u; v; w 2 V the integral in the definition of B is well defined, as from the Sobolev embedding theorem it follows that H 1 .˝/ L4 .˝/. Using this notation, Problem 5.1 can be equivalently reformulated as follows. Problem 5.2. Find u 2 V and 2 H.u / such that for every v 2 V we have hAu C B.u; u/; vi C .; v /W D hF; vi: We end this section by proving several auxiliary properties of the operators A and B. Lemma 5.3. The operator A W V ! V 0 defined by (5.17) is bilinear, continuous, and coercive. Moreover, we have hAu; ui D 2kuk2

for

u 2 V:

Proof. The assertion follows in a straightforward way from the definition of A and the definition of the norm in V, cf. (5.9). t u Lemma 5.4. For u; v; w 2 V we have hB.u; w/; vi cB kukkwkkvk;

(5.18)

hB.u; v/; vi D 0;

(5.19)

hB.u; w/; vi D hB.u; v/; wi:

(5.20)

Proof. We first prove (5.18). By the Hölder inequality hB.u; w/; vi kukL4 .˝/3 krwkL2 .˝/33 kvkL4 .˝/3 : From the Sobolev embedding theorem it follows that kzkL4 .˝/3 CkzkH 1 .˝/3 for all z 2 H 1 .˝/3 with a constant C > 0. Hence, we have hB.u; w/; vi C2 kukH 1 .˝/3 kwkH 1 .˝/3 kvkH 1 .˝/3 ; and (5.18) follows by the Korn inequality (5.8).

102


Q We have For the proof of (5.19), let u; v 2 V. 0 1 Z Z 3 X 1 hB.u; v/; vi D uj vi;j vi dx D uj @ vi2 A dx 2 ˝ ˝ jD1 D

1 2

Z ˝

uj;j

3 X

;j

! vi2

iD1

dx C

1 2

Z @˝

uj nj

3 X

vi2 dS D 0;

iD1

where the last equality follows from the fact that div u D 0 in ˝ and un D u n D 0 on @˝. Now as VQ is dense in V, for u; v 2 V we can construct sequences um ; vm 2 VQ such that kv vm k ! 0 and ku um k ! 0 as m ! 1. Hence hB.u; v/; vi D hB.u um C um ; v vm C vm /; v vm C vm i D hB.u um ; v/; vi C hB.um ; v vm /; vi C hB.um ; vm /; v vm i C hB.um ; vm /; vm i: We use the fact that hB.um ; vm /; vm i D 0 and (5.18) to get jhB.u; v/; vij cB .ku um kkvk2 C kv vm kkum k.kvk C kvm k//: The sequences um ; vm are bounded in V as they converge in this space, and the sequences ku um k and kv vm k converge to zero, whence the assertion is proved. Finally, (5.20) is a straightforward consequence of (5.19). For the proof it suffices to substitute v C w in place of v in (5.19). t u Lemma 5.5. The mapping V 2 3 .w; u/ ! B.w; u/ 2 V 0 is sequentially weakly continuous, i.e. wk ! w

and

uk ! u

weakly in V ) hB.wk ; uk /; vi ! hB.w; u/; vi

(5.21) for all

v 2 V:

Proof. First, take v 2 VQ and let uk ! u; wk ! w weakly in V. We will prove that hB.wk ; v/; uk i ! hB.w; v/; ui:

(5.22)

As the embedding V L2 .˝/3 is compact we have uk ! u and wk ! w strongly in L2 .˝/3 . Moreover, hB.wk ; v/; uk i hB.w; v/; ui D hB.wk ; v/; uk ui C hB.wk w; v/; ui; whence jhB.wk ; v/; uk i hB.w; v/; uij kvkC1 .˝/3 .kwk kL2 .˝/3 kuk ukL2 .˝/3 C kwk wkL2 .˝/3 kukL2 .˝/3 /;

5.4 Existence of Weak Solutions for the Case of Linear Growth Condition

103

whereas (5.22) is proved, and by (5.20) hB.wk ; uk /; vi ! hB.w; u/; vi:

(5.23)

Now, let v 2 V and let vm 2 VQ be a sequence such that kvm vk ! 0 as m ! 1. We have hB.wk ; uk /; vi D hB.wk ; uk /; vm i C hB.wk ; uk /; v vm i: From the fact that the sequences wk ; uk are bounded in V it follows that lim lim sup jhB.wk ; uk /; v vm ij lim lim sup cB kwk kkuk kkv vm k D 0;

m!1 k!1

m!1 k!1

which yields (5.21) and the proof is complete.

t u

5.4 Existence of Weak Solutions for the Case of Linear Growth Condition Define

(

2kc1 kL2 .C / C c2 k kkFkV 0 ; 2 c2 k k2 ) k kkc1 kL2 .C / C kFkV 0 kwk : 2 c2 k k2

S D .; w/ 2 W V W kkW

(5.24)

We formulate the main theorem of this section. Theorem 5.1. Assume .h1/–.h5/. Then Problem 5.1 has a solution .; u/ 2 S. Moreover, any solution of Problem 5.1 must belong to S. The proof of the above theorem will be done through several lemmas. First, we formulate the following auxiliary problem. Problem 5.3. Fix 2 W and w 2 V. Find u 2 V such that for every v 2 V we have hAu C B.w; u/; vi C .; v /W D hF; vi:

(5.25)

Lemma 5.6. For a given .; w/ 2 W V the auxiliary Problem 5.3 has exactly one solution u 2 V. Proof. Define the mapping M W V ! V 0 by M.u/ D Au C B.w; u/. Obviously this mapping is linear. By Lemma 5.3 and (5.18), M is bounded, and hence also continuous. Moreover, by Lemma 5.3 and (5.13) hMu; ui D hAu; ui D 2kuk2 ;

104


whence M is also coercive. Consider the functional G W V ! R defined as G.v/ D hF; vi .; v /W . Then G is linear and jG.v/j kFkV 0 kvk C kkW kv kW .kFkV 0 C kkW k k/kvk; whence G is bounded and, in consequence, it belongs to V 0 . By the Lax–Milgram lemma, it follows that Problem 5.3 has exactly one solution u 2 V. t u In view of the unique solvability of Problem 5.3, we can define the mapping 1 W W V ! V which assigns to the pair .; w/ 2 W V the unique solution u 2 V of (5.25). We can write hA1 .; w/ C B.w; 1 .; w//; vi C .; v /W D hF; vi:

(5.26)

Lemma 5.7. For every .; w/ 2 W V we have k1 .; w/k

k k kFkV 0 kkW C : 2 2

Proof. Taking the test function v D 1 .; w/ in (5.26) we get 2k1 .; w/k2 C .; 1 .; w/ /W D hF; 1 .; w/i: It follows that 2k1 .; w/k2 kkW k kk1 .; w/k C kFkV 0 k1 .; w/k; whence we have the assertion.

t u

Lemma 5.8. If k ! weakly in W, wk ! w weakly in V, and 1 .k ; wk / ! weakly in V, then D 1 .; w/, i.e., the graph of 1 is a sequentially closed set in .W; weak/ .V; weak/ .V; weak/ topology. Proof. Let k ! weakly in W, wk ! w weakly in V, and 1 .k ; wk / ! weakly in V. We have hA1 .k ; wk / C B.wk ; 1 .k ; wk //; vi C .k ; v /W D hF; vi; for v 2 V. We pass to the limit in all terms in the last equation. As A is linear and continuous, it is also weakly sequentially continuous, and we have hA1 .k ; wk /; vi ! hA; vi: Lemma 5.5 implies that hB.wk ; 1 .k ; wk //; vi ! hB.w; /; vi;

5.4 Existence of Weak Solutions for the Case of Linear Growth Condition

105

and weak convergence k ! in W implies that .k ; v /W ! .; v /W : Summarizing, we get hA C B.w; /; vi C .; v /W D hF; vi; whereas D 1 .; w/ and the proof is complete. We define the multifunction W W V ! 2

WV

t u by the formula

.; w/ D H.w / f1 .; w/g: We will show that satisfies all assumptions of the Kakutani–Fan–Glicksberg fixed point theorem, cf. Theorem 3.8, and, in consequence, has a fixed point .; w/. We observe that every fixed point of must be a solution of Problem 5.1. Indeed, assume that .; w/ is a fixed point of . Then 2 H.w / and w D 1 .; w/. From the definition of 1 it follows that hAw C B.w; w/; vi C .; v /W D hF; vi for all v 2 V: Clearly, .; w/ solves Problem 5.1. Hence, be showing the existence of a fixed point of we prove the existence of a solution for Problem 5.1. Lemma 5.9. Assume .h1/–.h5/. Then .S/ S and if x 2 .x/, then x 2 S. Proof. First we prove the assertion that .S/ S. Let .; u/ 2 .; w/, where .; w/ 2 S. Then 2 H.w / and u D 1 .; w/. From Lemma 5.7 it follows that kuk

k k kFkV 0 kkW C : 2 2

(5.27)

By Lemma 5.1 it is possible to use the growth condition .H3/. We get kkW kc1 kL2 .C / C c2 k kkwk:

(5.28)

As .; w/ 2 S, we can use the bounds given in the definition (5.24). The estimate (5.27) implies that kuk

k kkc1 kL2 .C / C kFkV 0 k k 2kc1 kL2 .C / C c2 k kkFkV 0 kFkV 0 D C ; 2 2 2 c2 k k 2 2 c2 k k2

and the estimate (5.28) implies that kkW kc1 kL2 .C / Cc2 k k

k kkc1 kL2 .C / C kFkV 0 2kc1 kL2 .C / C c2 k kkFkV 0 D : 2 c2 k k2 2 c2 k k2

We have proved that u and satisfy the estimates given in definition (5.24) of the set S, whence .; u/ 2 S. The proof of the first assertion is complete.

106


For the proof of the second assertion assume that .; u/ 2 .; u/. We must show that .; u/ 2 S, i.e., .; u/ satisfy the bounds given in definition (5.24). By Lemma 5.7 and the growth condition .H3/ in Lemma 5.1 we have kuk

k k kFkV 0 kkW C ; kkW kc1 kL2 .C / C c2 k kkuk: 2 2

Combining the above two inequalities in two different ways we obtain k k kFkV 0 .kc1 kL2 .C / C c2 k kkuk/ C ; 2 2 k k kFkV 0 kkW C : kkW kc1 kL2 .C / C c2 k k 2 2 kuk

A straightforward calculation proves that .; u/ satisfy the bounds given in the definition (5.24), whereas indeed .; u/ 2 S. The proof is complete. u t The space W V endowed with the .W; weak/ .V; weak/ topology is a locally convex Hausdorff topological vector space, and the set S is nonempty, compact, and convex in this space. Assuming .h1/–.h4/, the values of are nonempty and convex (see .H1/ of Lemma 5.1). Lemma 5.10. Assume .h1/–.h5/. Then the graph of is sequentially closed in .W; weak/ .V; weak/ .W; weak/ .V; weak/. Proof. Let k ! weakly in W and wk ! w weakly in V. Let moreover .k ; uk / be such that k ! weakly in W, uk ! u weakly in V and .k ; uk / 2 .k ; wk /. The last assertion means that k 2 H.wk / and uk 2 1 .k ; wk /. Lemma 5.8 implies that u 2 1 .; w/. Since the trace operator W V ! W is compact, we have wk ! w strongly in W. It remains to use .H2/ of Lemma 5.1 to conclude that 2 H.w /, whence .; u/ 2 .; w/. The proof is complete. t u Lemma 5.11. Assume .h1/–.h5/. Then the graph of jS is a closed set in topology .W; weak/ .V; weak/ .W; weak/ .V; weak/. Proof. Lemma 5.10 implies that the graph of jS is sequentially closed in topology .W; weak/ .V; weak/ .W; weak/ .V; weak/. The graph of jS is contained in S S, a set which is compact, and, by the Eberlein–Šmulyan theorem also sequentially compact in .W; weak/ .V; weak/ .W; weak/ .V; weak/ topology. Thus the graph of jS must be sequentially compact, and by the Eberlein–Šmulyan theorem, it is also compact and hence closed. t u We have all the ingredients to prove that main theorem of this section. Proof (of Theorem 5.1). The assertion of the theorem is a straightforward application of the Kakutani–Fan–Glicksberg fixed point theorem and Lemmata 5.9 and 5.11. t u

5.5 Existence of Weak Solutions for the Case of Power Growth Condition

107

5.5 Existence of Weak Solutions for the Case of Power Growth Condition As in this section we no longer impose the linear growth condition .h4/, but we replace it with the more general power growth condition .h6/, we cannot interpret R anymore the term C v dS in (5.15) as the scalar product in W. Instead, we represent this term using the duality between U and U 0 , and the multivalued operator G given by (5.7). We are in position to formulate the following problem. Problem 5.4. Find u 2 V and 2 G.u / such that for every v 2 V we have hAu C B.u; u/; vi C h; v iU0 U D hF; vi:

(5.29)

The main result of this section is the following existence theorem for the above problem. Theorem 5.2. Assume .h1/–.h3/ and .h6/, .h7/. Then Problem 5.4 has a solution .; u/ 2 W V. Proof. The proof consists of two steps. In the first step we will define the cut-off friction multifunction hm and the associated cut-off Problem 5.5. In the second step we pass to infinity with the cut-off constant and we recover the solution of the original problem. Step 1. Let us first choose a natural number m and define 8
hm .x; s/ D ms :h x; jsj if jsj > m: We will show that hm satisfies .h1/–.h5/. Naturally, hm satisfies .h1/ and .h2/. We will prove that hm satisfies .h3/, i.e., the graph of s ! hm .x; s/ is closed in R6 for a.e. x 2 C . To this end, let sk ! s and k ! with k 2 hm .x; sk /. Define ( sk if jsk j m; rk D ms k if jsk j > m: jsk j From the fact that k 2 hm .x; sk / and the definition of hm it follows that k 2 h.x; rk /. We will consider two cases. If jsj m, then we have rk ! s and we use .h3/ for h whence it follows that 2 h.x; s/ D hm .x; s/ for a.e. x 2 C . In the second case and we can use .h3/ for h to deduce we have jsj > m. Then we have rk ! ms jsj

ms that 2 h x; jsj D hm .x; s/ for a.e. x 2 C . The proof that hm satisfies .h3/ is complete. We will now demonstrate that hm satisfies .h4/ and .h5/. Indeed, let s 2 R3 and 2 hm .x; s/. By the definition of hm and .h6/ for a.e. x 2 C we have jj c3 .x/ C c4 mp1 , whence .h4/ holds with c1 D c3 ./ C c4 mp1 2 L2 .C / and c2 D 0, which implies that hm satisfies also .h5/.

108


We have proved that hm satisfies .h1/–.h5/. We can define the multivalued operator Hm W W ! 2W by the formula Hm .u/ D f 2 W W .x/ 2 hm .x; u.x//

a.e. on

C g:

We formulate the following cut-off version of Problem 5.4, where the cut-off parameter is denoted by m. Problem 5.5. Find um 2 V and m 2 Hm .um / such that for every v 2 V we have hAum C B.um ; um /; vi C .m ; v /W D hF; vi:

(5.30)

As hm satisfies .h1/–.h5/ we can use Theorem 5.1 to deduce that for each m there exists .m ; um / 2 W V, a solution of Problem 5.5. Step 2. We pass with the cut-off constant m to infinity and we recover the solution of Problem 5.4. Taking v D um in (5.30) we get 2kum k2 C .m ; um /W kFkV 0 kum k:

(5.31)

Now we use .h7/ to obtain Z .m ; um /W D m um dS C

Z D fjum jmg

Z

m um dS C

Z

fjum j>mg

Z

2

fjum jmg

d1 jum j C d2 dS

d1 kum k2W kd2 kL1 .C /

jum j mum m dS m jum j

fjum j>mg

jum j .d1 m2 C d2 / dS m

1 kd2 kL2 .C / kum kW : m

Using the last inequality in (5.31) we get, by the trace theorem .2 d1 k k2 /kum k2 kFkV 0 kum k C kd2 kL1 .C / C

1 k kkd2 kL2 .C / kum k: m

It follows that the sequence um is bounded in V and thus, for a subsequence, denoted by the same index, um ! u weakly in V. From the fact that p 2 Œ2; 4/ it follows that the trace operator p W V ! U is compact (see, for example, Theorem 2.79 in [50]) and um ! u strongly in U. Using .h6/ we estimate Z q km kU0

Z

fjum jmg

.c3 C jum j

/ dS C

p1 q

Z

q

C

2q1 .c3 C jum jp / dS:

fjum j>mg

ˇ !q ˇ ˇ mum ˇp1 ˇ dS c3 C ˇˇ jum j ˇ


109

Since c3 2 L2 .C / Lq .C / and the sequence um which converges in U strongly is norm bounded in this space, it follows that m is bounded in U 0 and hence, for a subsequence, not renumbered, m ! weakly in U 0 . We can rewrite (5.30) as hAum C B.um ; um /; vi C hm ; v iU0 U D hF; vi; and we can pass to the limit to obtain (5.29). We only need to show that 2 G.u /. Since um ! u strongly in U, then, for a subsequence, denoted by the same index, um .x/ ! u .x/ with jum .x/j g.x/ for a.e. x 2 C , where g 2 Lp .C /. Fix " > 0. By the Luzin theorem there exists a closed set M" C , with m2 .C n M" / < " and g continuous on M" . Define m" D maxx2M" g.x/. For all natural m m" we have jum .x/j g.x/ m" m a.e. on M" and hence m .x/ 2 h.x; um .x// a.e. on M" . We can use .H2/ of Theorem 3.23 for the space Z D Lp .M" /3 and the map Z 3 u ! f 2 Z 0 W .x/ 2 h.x; u.x//

a.e. on

M" g 2 2Z

0

(indeed um ! u strongly in Z and m ! weakly in Z 0 ), whereas a.e. on M" we have .x/ 2 h.x; u .x//. Since " > 0 was arbitrary, it follows that .x/ 2 h.x; u .x// a.e. on C and the assertion is proved. This completes the proof of Theorem 5.2. u t

5.6 Comments and Bibliographical Notes In this chapter the multivalued term has the form of an arbitrary multifunction. In most typical applications this multifunction is a Clarke subdifferential of a certain locally Lipschitz functional, usually nondifferentiable in a discrete or Lebesgue null set of points. In such a case, the multivalued PDE is equivalent to a certain hemivariational inequality, as it has been proved by Carl [49] (the proof in [49] is done for an evolution problem but it works well for the static case, too). There are many techniques to prove existence of weak solutions for the multivalued elliptic partial differential equations or hemivariational inequalities. The idea based on the use of surjectivity results for pseudomonotone operators has been extensively exploited in [184]. The application of this technique to stationary incompressible Navier–Stokes problem with the multivalued feedback control boundary condition in the normal direction involving the Bernoulli pressure was presented in [175]. The approach based on the Galerkin method and the finite element method was studied in [114], and the techniques using upper and lower solutions were thoroughly analyzed in [50]. Yet another interesting approach to study the existence of weak solutions using the Kuratowski–Knaster–Mazurkiewicz (KKM) principle was proposed in [78]. The approach based on the Kakutani–Fan–Glicksberg fixed point theorem is, to our knowledge, new.

110


Fig. 5.2 Two examples of a frictional contact condition. The vertical line in both plots represents the stick effect, where the tangential velocity of the fluid u is equal to the given velocity U0 . In such a case the friction condition is actually a (nonhomogeneous) Dirichlet condition. If the stick does not occur, i.e., u ¤ U0 we have a slip effect. In such a case, the tangential stress T is given as a function of u . The right plot depicts the possibility of taking into account the dependence of the friction coefficient of the slip rate. It is a priori unknown which part of the contact boundary is a stick zone, and which part is the slip zone. Of course, in the case of evolution problems, both zones can change with time

The book [177] is devoted to the applications of multivalued PDEs in the continuum mechanics and presents many examples of contact conditions in the solid mechanics leading to multivalued PDEs and hemivariational inequalities. An example of the law that can be used in the model presented in this chapter is the Tresca friction law 8n o < k u U0 for u ¤ U0 ; ju U0 j h.x; u / D :B.0; k / otherwise: The interpretation of this law is the following: on the contact boundary the foundation is moving with the given velocity U0 2 R3 . If there is no slip, i.e., u D U0 then the magnitude of the friction force cannot exceed the friction bound k . When the friction force is large enough, its absolute value equal to k , then the slip appears and we have u ¤ U0 . The direction of the friction force is always opposite to the vector u U0 . The Tresca law is presented in the left plot in Fig. 5.2. Obviously, this example satisfies all the assumptions .h1/–.h5/, as we have .h4/ with c2 D 0. It is possible to generalize the Tresca law, making the friction bound k dependent on ju U0 j. In such a case we speak of slip rate dependent friction. In particular the situation when k is decreasing with increasing ju U0 j reflects the fact that the kinetic friction is less than the static friction. This, generalized, version of the Tresca law is presented in the right plot in Fig. 5.2. The dependence of the k on ju U0 j can also be discontinuous, to account for flattening of asperities on the surface. Similar laws were used, for example, in [12, 213] for static problems in elasticity, see Fig. 4.6 in [12] and Fig. 2a in [213] for examples of particular nonmonotone friction laws.

6

Stationary Flows in Narrow Films and the Reynolds Equation

In this chapter we study a typical problem from the theory of lubrication, namely, the Stokes flow in a thin three-dimensional domain ˝ " , " > 0. We assume the Fourier boundary condition (only the friction part) at the top surface and a nonlinear Tresca interface condition at the bottom one. When " ! 0 the domains ˝ " become in the limit a two-dimensional domain !, the mutual bottom of the domains ˝ " . Let .u" ; p" / be the solution of the considered Stokes problem in ˝ " . An important problem in the theory of lubrication, which deals with flows in thin films, is to describe the behavior of solutions .u" ; p" / as " ! 0 in terms of some limit functions .u? ; p? /. It happens that the pressure p? depends only on two variables which define the surface ! and satisfies the Reynolds equation, a fundamental equation in this theory. We can say that the Reynolds equation describes a distribution of the pressure “in an infinitely thin domain.” Moreover, the boundary conditions imposed on the solutions of the Stokes equation in ˝ " tend in an appropriate sense to some boundary conditions for the Reynolds equation. Our aim is to provide a strict mathematical analysis of this asymptotic problem.

6.1 Classical Formulation of the Problem We consider a boundary value problem for the Stokes equations in ˝ " , u C rp D 0 in

˝ ";

(6.1)

div u D 0 in

˝ ";

(6.2)


111

112

6 Stationary Flows in Narrow Films and the Reynolds Equation

The domain ˝ " is as follows. Let ! R2 be an open bounded set with sufficiently smooth boundary. Then ˝ " D f.x1 ; x2 ; x3 / 2 R3 W

.x1 ; x2 / 2 !;

0 < x3 "h.x1 ; x2 /g:

The boundary of ˝ " consists of three parts: the bottom !, the lateral part L" , and the top surface F" , @˝ " D ! [ L" [ F" . Our boundary conditions are: • At the bottom ! we assume un D u n D 0

!:

on

(6.3)

The tangential velocity on ! is unknown and satisfies the Tresca boundary condition with the friction coefficient k" , jT j D k" H) 9 0 u D s T ; jT j < k" H) u D s;

(6.4)

where j:j denotes the Euclidean norm in R2 , and s is the velocity of the lower surface !, n D .n1 ; n2 ; n3 / is the outward unit normal to ˝ " , and un D u n;

u D u un n:

Boundary condition (6.3)–(6.4) is a form of the well-known Coulomb solid– solid interface law [91] applied here for the fluid–solid interface. Another variant, taking into account also the normal stresses, was considered in [21]. • On the lateral boundary L" we assume u D G"

on

L" :

(6.5)

• At the top surface F" we assume the Fourier boundary condition T C l" u D 0 on

F" ;

(6.6)

and unD0

on

F" ;

F"

given by

x3 D h" .x1 ; x2 / D "h.x1 ; x2 /:

(6.7)

Two-dimensional cross section of the domain of the studied problem is presented in Fig. 6.1. We have to specify the data G" in (6.5), l" in (6.6), and h" in (6.7), depending on ". We will specify this dependence in Sect. 6.3.

6.1 Classical Formulation of the Problem

113

We assume that G" 2 .H 1 .˝ " //3 with div G" D 0 in ˝ " , G" n D 0 on F" [ !. We assume also that h satisfies 0 < h.x/ hM ;

h 2 C2 .!/;

(6.8)

and that l" > 0. In (6.4) and (6.6) T is the tangential part of the stress vector Tn on the boundary @˝ " , where the stress tensor T is given by Tij D p ıij C 2Dij .u/

where

Dij .u/ D

1 2

@uj @ui C @xj @xi

for

1 i; j 3;

Fig. 6.1 Two-dimensional cross section of the domain of the studied problem. For " D 1 we denote F" D F and L" D L . On F we assume the Fourier condition, on L , the (nonhomogeneous) Dirichlet condition, and on !, the Tresca condition

namely T D Tn Tn n;

(6.9)

where Tn D .Tn/ n is the normal part of the stress tensor on the boundary @˝ " . Remark 6.1. In [15, 21] a similar problem was considered, it differed from the above in a boundary condition at the top surface; namely, in these papers it was assumed that u D 0 at

F" :

(6.10)

Condition (6.10) helps much in obtaining the solution. With (6.6) and (6.7) in place of (6.10) the problem is more difficult due to the lack of the usual forms of the Poincaré and Korn inequalities. What is important is to obtain other forms of these inequalities, in which the constants do not explode as " tends to zero. The plan of the rest of the chapter is as follows. In Sect. 6.2 we provide a weak formulation of the problem, prove suitable forms of the Korn and Poincaré inequalities, and obtain key estimates. In Sect. 6.3, taking into account a small parameter, we introduce a scaling, and using estimates from Sect. 6.2 prove a convergence theorem for the rescaled functions. In Sect. 6.4 we establish the limit variational inequality, give precise characterization of the sets to which its solution

114


and admissible test functions belong, and prove the strong convergence of the velocity fields. In Sect. 6.5 the Reynolds equation and the limit boundary conditions are obtained. In the last section we prove the uniqueness of solutions of the limit problem.

6.2 Weak Formulation and Main Estimates We start from remarks on the notation in this chapter. We will denote by x D .x1 ; x2 / variables belonging to the set !. Surface element in integrals will be denoted by dS, or, in case of integrals over ! by dx or dx1 dx2 . The real variable in the vertical direction, orthogonal to ! will be denoted by y or x3 and the corresponding length element in integrals by dy or dx3 . Volume element in volume integrals will be denoted by dx dy or dx dx3 . Let us define K D f' 2 .H 1 .˝ " //3 W

' G" D 0 on

L" ;

'nD0

on F" [ !g; (6.11)

where G" 2 .H 1 .˝ " //3

with div G" D 0 in

˝ ";

G" n D 0

on

1" [ !; (6.12)

div v D 0 in ˝ " g;

Kd D fv 2 K W and L02 .˝ " /

2

"

Z

D fp 2 L .˝ / W

˝"

p dx dx3 D 0g:

Our aim is to justify the following weak formulation of problem (6.1)–(6.7). Definition 6.1. The weak solution of problem (6.1)–(6.7) is a pair .u; p/ such that u 2 Kd ; p2

Z a.u; ' u/

˝"

Z C

!

(6.13)

L02 .˝ " /;

p div ' dx dx3 C

Z F"

l" u.' u/ dS

k" .j' sj ju sj/ dx 0 for all ' 2 K;

(6.14)

where Z a.u; v/ D 2

˝"

Dij .u/Dij .v/ dx dx3 :

(6.15)

6.2 Weak Formulation and Main Estimates

115

Remark 6.2. In (6.14), k" comes from the Tresca condition. It is a positive constant and its dependence on " shall be defined in Sect. 6.3. Observe that from condition (6.13) it follows that: u D G" unD0

on

L"

on

F" [ !

.cf: (6.5)/;

.cf: (6.3)

and

(6.7)/:

Tresca and slip conditions are included in (6.14). To obtain (6.14) we observe that ui C p;i D 2.ui;j C uj;i / C pıij ;j Tij;j ; as uj;ij D .uj;j /;i D 0 from div u D 0. We multiply the equations of motion Tij;j D 0

.i D 1; 2; 3/

by 'i ui , integrate by parts in ˝ " and add for i D 1; 2; 3, to get Z

@.'i ui / Tij dx dx3 @xj ˝"

Z @˝ "

Tn .' u/ dS D 0:

(6.16)

Now Z ˝"

Tij

@.'i ui / dx dx3 D @xj

Z ˝"

Tij Dij .'i ui / dx dx3 ;

since Tij Dij .v/ D

1 1 1 Tij .vi;j C vj;i / D Tij vi;j C Tji vj;i D Tij vi;j 2 2 2

by symmetry. From (6.16) and (6.15), Z ˝"

Tij

@.'i ui / dx dx3 D @xj

Z ˝"

.2Dij .u/ p ıij /Dij .' u/ dx dx3 Z

D a.u; ' u/ Z D a.u; ' u/ as div u D 0 in ˝ " .

˝"

˝"

p div .' u/ dx dx3 p div ' dx dx3 ;

(6.17)

116


The second term in (6.16) is divided into three parts, as @˝ " D F" [ ! [ L" : We have, by (6.9) and (6.6), Z Z ŒT C Tn n ..' u/ C .' u/n n/ dS Tij nj .'i ui / dS D F"

Z D Dl

"

Z

F"

F"

F"

.l" u/ .' u/ dS u .' u/ dS;

as .' u/n D n .' u/ D 0 on F" by (6.11) and (6.12). Integration over ! and the Tresca condition yield Z Z k" .j' sj ju sj/ dx Tn .' u/ dx; !

(6.18)

!

where s D .s1 ; s2 ; 0/ D G" j! :

(6.19)

Observe that G" n D 0 on ! by (6.12) so that G"3 D 0: The third integral is Z Tn .' u/ dS D 0; 1 L"

as ' D u D G" on L" by (6.11) and (6.13). In this way, from (6.16)–(6.18) we get Z a.u; ' u/

˝"

p div ' dx dx3 C l

"

C k"

Z F"

u.' u/ dS

(6.20)

Z

!

.j' sj ju sj/ dx 0:

Remark 6.3. Observe that in a.:; :/ there is the Newtonian viscosity . In the sequel we set D 1. We use (6.20) to obtain the basic estimate of the velocity given in Lemma 6.4 below. Lemma 6.1. We have Z Z Z Z 1 l" l" a.u; u/ C juj2 dS C k" ju sj dx jG" j2 dS C 2 jrG" j2 dx dx3 : " " 2 2 F" 2 ! F ˝ (6.21)


117

Proof. We set ' D G" in (6.20) to obtain, a.u; u/ C l"

Z F"

juj2 dS C k"

Z !

ju sj dx a.u; G" / C l"

Z F"

uG" dS;

(6.22)

as div G" D 0, and G" D s on ! by (6.19). Moreover, as D 1, Z

"

a.u; G / D 2 Z 2

˝"

D

Dij .u/Dij .G" / dx dx3 1

˝"

Z

˝"

1

.Dij .u/Dij .u// 2 .4Dij .G" /Dij .G" // 2 dx dx3 Z

Dij .u/Dij .u/ dx dx3 C 4

˝"

Dij .G" /Dij .G" // dx dx3

1 1 a.u; u/ C 2a.G" ; G" / a.u; u/ C 2 2 2

Z ˝"

jrG" j2 dx dx3 : (6.23) t u

From (6.22) and (6.23) we obtain immediately (6.21).

The desired estimate of the velocity will follow from (6.21) and a form of the Korn inequality, suitable for our problem. Lemma 6.2 (A Form of the Korn Inequality). We have Z Z jr.u G" /j2 dx dx3 a.u G" ; u G" / C C.F" / ˝"

F"

ju G" j2 dS;

(6.24)

where

C.F" / D 2kD2 h" kC.!/ 1 C kD1 h" k2C.!/ :

(6.25)

Observe that we use here h 2 C2 .!/, cf. (6.8), and that C.F" / is of order " if h" D "h as in (6.7). Proof. Assume that v D u G" is a smooth function. The result will follow from density of smooth functions in Kd . We have a.v; v/ D

1 2 Z

Z ˝"

@vi @vk C @xk @xi

2 dx dx3

@vi @vi @vi @vk dx dx3 D C @xk @xk @xk @xi ˝" Z Z @vi @vk 2 D jrvj dx dx3 C dx dx3 " " @x k @xi ˝ ˝

118


Z D Z D Z

Z

2

˝"

˝"

jrvj dx dx3 jrvj2 dx dx3 C

@˝ "

@vi vk nk dS C @xi

If div v D 0, then we have Z Z jrvj2 dx dx3 D a.v; v/ C ˝"

˝"

@2 vi vk dx dx3 C @xk @xi

˝"

@vi @vk dx dx3 @xi @xk

@˝ "

@vi vk ni dS: @xk

Z Z

@˝ "

@vi vk nk dS @xi

Z @˝ "

Z @˝ "

@vi vk ni dS @xk

@vi vk ni dS: @xk

(6.26)

For v D uG" we have .uG" /n D 0 on @˝ " as u 2 Kd and G" is as in (6.12) thus the first surface integral on the right-hand side equals zero, and the second surface integral reduces to that over F" [ !; as v D u G" D 0 on L" . We have to estimate the integral Z F" [!

@.ui G"i / .uk G"k /ni dS D @xk

Z

Since .u G" / n D 0 on F" [ !, then 3 X @.ui G" / i

iD1

@xk

F" [!

.uk G"k /

@.uG" /n @xk

ni D

3 X

@.ui G"i / ni dS: @xk

(6.27)

D 0, i.e.,

.ui G"i /ni;k :

(6.28)

iD1

But ni;k D 0 on !, so that from (6.27) and (6.28) we have ˇZ ˇ Z ˇ ˇ " G / @.u i ˇ ˇ Q " i " ni dSˇ C.F / ju G" j2 dS; ˇ .uk Gk / ˇ F" ˇ @xk F"

(6.29)

where Q F" / 2 max jni;k .q/j: C. " q2F

As F" is given by x3 D h" .x1 ; x2 /,

" @h" @h .x ; x /; .x ; x /; 1 1 2 1 2 @x1 @x2 D n.x1 ; x2 /: n.q/ D p 1 C jrh" .x/j2

(6.30)


119

We compute n3;i .x1 ; x2 / D

@n3 3 .x1 ; x2 / D .1 C jrh" j2 / 2 @xi

@h" @2 h" @h" @2 h" C @x1 @xi @x1 @x2 @xi @x2

;

whence ˇ ˇ ˇ @n3 ˇ " " " " 2 ˇ ˇ ˇ @x ˇ 2jD1 h jjD2 h j jD2 h j.1 C jD1 h j /; i and similarly ˇ ˇ ˇ @nj ˇ jnj;i .x1 ; x2 /j D ˇˇ ˇˇ jD2 h" j.1 C jD1 h" j2 /: @xi Q F" / C.F" /, where C.F" / is defined in (6.25). In the end, from (6.26) Thus C. and (6.29) we obtain (6.24). t u R Now, having inequality (6.24), we can estimate ˝ " jruj2 dx dx3 . We have C.F" /

Z

" 2

F"

ju G j dS

2C.F" /

Z F"

!

Z

2

juj dS C

" 2

1"

jG j dS ;

(6.31)

and, by (6.23), a.u G" ; u G" / D a.u; u/ C a.G" ; G" / 2a.u; G" / Z 2a.u; u/ C 5 jrG" j2 dx dx3 :

(6.32)

˝"

Moreover Z Z jruj2 dx dx3 2 ˝"

˝"

jr.u G" /j2 dx dx3 C 2

Z ˝"

jrG" j2 dx dx3 :

(6.33)

In the end, from (6.24), along with (6.31)–(6.33) we have Lemma 6.3. Z Z 2 jruj dx dx3 4 a.u; u/ C 12 ˝"

C

˝"

jrG" j2 dx dx3

4C.F" /

Z

2

F"

juj dS C

(6.34) !

Z

" 2

F"

jG j dS :

The first term on the right-hand side as well as the surface integral estimated in (6.21). From (6.21) and (6.34) we have

R

F"

juj2 dS is

120


Lemma 6.4. Z

Z 16C.F" / jruj dx dx3 28 C jrG" j2 dx dx3 l" ˝" ˝" Z C 4 l" C 2C.F" / jG" j2 dS: 2

(6.35)

F"

Remark 6.4. If we set "l" D Ol 2 .0; 1/, then l" ! 1 as " ! 0, and 16C.F" / D O.1/ as " ! 0; l" 1 " " as " ! 0: 4.l C 2C.F // D O " 28 C

R Later on we shall show that " ˝ " jruj2 dx dx3 D O.1/ as " ! 0. This will come from estimates of the integrals on the right-hand side of (6.35), by means of the H 1 .˝/3 norm of a given function GO in H 1 .˝/3 , to which the family of functions G" is suitably related (here andRin the sequel we put ˝ D ˝ 1 ). To be able to estimate ˝ juj2 dx dy we need a suitable form of the Poincaré inequality. Lemma 6.5 (A Form of the Poincaré Inequality). Set h"M D max! jh" .x1 ; x2 /j: Then Z ˝"

juj2 dx dx3 2h"M

Z F"

ˇ ˇ ˇ @u ˇ2 ˇ ˇ dx dx3 : ˇ ˇ ˝ " @x3

Z

juj2 dS C 2.h"M /2

(6.36)

Proof. Let .x1 ; x2 ; x3 / D .x; t/, x D .x1 ; x2 / for short. Then u.x; t/ D u.x; h" .x//

Z

h" .x/ t

@u .x; z/ dz; @z

so that "

2

"

ju.x; t/j 2ju.x; h .x//j C 2h .x/

Z

h" .x/ 0

ˇ2 ˇ ˇ ˇ @u ˇ .x; z/ˇ dz: ˇ ˇ @z

We integrate over t 2 Œ0; h" .x/ to obtain Z 0

h" .x/

2

"

"

2

"

ju.x; x3 /j dx3 2h .x/ju.x; h .x//j C 2.h .x//

and, after integration over !, we obtain (6.36).

2

Z 0

h" .x/

ˇ ˇ2 ˇ @u ˇ ˇ .x; x3 /ˇ dx3 ; ˇ @z ˇ t u

6.3 Scaling and Uniform Estimates

121

We have now all basic estimates. In Sect. 6.3 we shall state them once again but in new variables: uO instead of u" , etc., defined in ˝ 1 . We have the following existence theorem in ˝ " . Theorem 6.1. For the data as in Sects. 6.1 and 6.2, there exists a unique weak solution .u; p/ D .u" ; p" / of problem (6.1)–(6.7), u" 2 Kd

and

p" 2 L02 .˝ " /:

Proof. For the proof, which requires some more advanced variational methods to be presented in this place, we refer the reader to [15] where the functional j is here given by Z j.'/ D !

k" j'.x; 0/ sj dx C l"

Z F"

u' dS:

t u

6.3 Scaling and Uniform Estimates Let ˝ D ˝ 1 and define, for a velocity u in ˝ " , yD

x3 ; "

uO i .x; y/ D ui .x; x3 /;

i D 1; 2

and

uO 3 .x; y/ D

1 u3 .x; x3 /; "

uO .x; y/ D .Ou1 .x; y/; uO 2 .x; y/; uO 3 .x; y// being a new velocity in ˝. For the pressure p in ˝ " we define pO .x; y/ D "2 p.x; x3 /: Let Ol D "l"

and

kO D "k" ;

where Ol and kO are some positive numbers, l" stands in (6.6), and k" appears in the Tresca condition (6.4). O y/ D .GO 1 .x; y/; GO 2 .x; y/; GO 3 .x; y// in H 1 .˝/3 such Now, for a given function G.x; O that G n D 0 on F [ ! .F D F1 / and @GO 1 @GO 2 @GO 3 D 0 .div GO D 0/; C C @x1 @x2 @y we define a family of functions G" in ˝ " , by GO i .x; y/ D G"i .x; x3 /;

i D 1; 2;

1 GO 3 .x; y/ D G"3 .x; x3 /: "

122


By KO we define the set n KO D ' 2 .H 1 .˝//3 W

' GO D 0 on

L ;

' n D 0 on

o F [ ! ;

where L D L1 . Observe that div GO D 0 in ˝ implies div G" D 0 in ˝ " . It is clear that uO D GO on L implies u D G" on L" , and uO n D 0 on ! implies u n D 0 on !. Moreover, from (6.30) it follows that also uO n D 0 on F implies u n D 0 on F" . We have proved in fact that O div uO D 0: is equivalent to uO 2 K;

u 2 Kd

To compare solutions .u" ; p" / for various " (these solutions exist by Theorem 6.1) we transform them as above to the same domain ˝. The transformed solutions .Ou" ; pO " / are solutions of a suitably transformed problem from Definition 6.1. We shall write the problem precisely later on. Now, we obtain estimates for uO D uO " directly from those obtained for u D u" in Sect. 6.2. We shall write estimates (6.35) and (6.36) in new variables. Lemma 6.6. We have Z Z 1 " 2 O 2 dx dy for all jrG j dx dx3 jr Gj " ˝ ˝"

" 2 .0; 1:

(6.37)

Exercise 6.1. Prove (6.37). Lemma 6.7. We have Z Z O 2 C jr Gj O 2 dx dy jGj jG" j2 dS C0 .˝/ F"

for all

˝

" 2 .0; 1;

(6.38)

where C0 .˝/ is independent of ". Proof. For i D 1; 2 we have Z F"

jG"i j2

Z dS D Z

!

!

p jGi .x; h" .x//j2 1 C jrh" .x/j2 dx Z p jGO i .x; h.x//j2 1 C jrh" .x/j2 dx D

F

jGO i j2 dS; (6.39)

6.3 Scaling and Uniform Estimates

123

(by definition Gi .x; h" .x// D GO i .x; h.x//), and Z F"

jG"3 j2 dS D

Z !

p jG"3 .x; h" .x//j2 1 C jrh" .x/j2 dx

p p 1 C jrh" .x/j2 2 2 O " jG3 .x; h.x//j 1 C jrh.x/j p dx 1 C jrh.x/j2 ! Z p 2 " 2 " max 1 C jrh .x/j jGO 3 j2 dS: (6.40) 2

Z

x2!

F

From (6.39) and (6.40), Z F"

jG" j2 dS

Z

O 2 dS: jGj

(6.41)

F

As Z F

O 2 dS C0 .˝/ jGj

Z O 2 C jr Gj O 2 dx dy; jGj

(6.42)

˝

where C0 .˝/ depends only on ˝, (6.41) and (6.42) give (6.38).

t u

Using inequalities (6.37) and (6.38) in (6.35) and taking into account Remark 6.4 we see that Z Z 2 O 2 C jr Gj O O 2 dx dy CQ 0 for all " 2 .0; 1: " jGj jruj dx dx3 C.˝; h; l/ ˝"

˝

(6.43)

Writing (6.43) in terms of uO , we obtain "2

2 X @Oui 2 @x 2

i;jD1

j

L .˝/

C

2 X @Oui 2 @y 2 iD1

L

2 X @Ou3 2 @Ou3 2 4 C "2 C " CQ 0 ; @y 2 @x 2 i L .˝/ L .˝/ .˝/ iD1 (6.44)

and also, as in [15, 21], @Op C1 @x 1 i H .˝/

@Op "C2 : @y 1 H .˝/

and

Exercise 6.2. Prove estimates (6.45). First, for in (6.14), where u D u" , p D p" , to get a.u" ; /

Z ˝"

p" div

(6.45)

2 H01 .˝ " / set ' D u ˙

dx dx3 D 0:

124


Further estimates follow from the Poincaré variables, Z Z Z jOui j2 dx dy 2hM jOui j2 dS C 2h2M ˝

inequality (6.36) written in new

ˇ ˇ2 ˇ @Oui ˇ ˇ ˇ dx dy ˇ ˇ ˝ @y

F

for

i D 1; 2; 3: (6.46)

Observe that changing the variables, we obtain the same form of the Poincaré inequality. Exercise 6.3. Prove (6.46). To estimate the integrals on the left-hand side, we have to estimate the surface integrals on the right-hand side. We have, Z

Z

2

F

jOui j dS D

!

p jOui .x; h.x//j2 1 C jrh.x/j2 dx

p p 1 C jrh.x/j2 D jOui .x; h .x//j 1 C jrh" .x/j2 p dx 1 C jrh" .x/j2 ! Z p max 1 C jrh.x/j2 jui j2 dS for i D 1; 2; (6.47) Z

"

2

F"

x2!

and Z

Z

2

F

jOu3 j dS D

!

p jOu3 .x; h.x//j2 1 C jrh.x/j2 dx

p 1 max 1 C jrh.x/j2 2 " x2!

Z F"

ju3 j2 dS:

(6.48)

From (6.21) using (6.37) and (6.38) we have Z F"

juj2 dS

Z F"

jG" j2 dS C

C0 .˝/

Z ˝

4 l3

Z ˝"

jrG" j2 dx dx3

O 2 C jr Gj O 2 / dx dy C .jGj

O C.˝; Ol; G/:

4 Ol

Z

O 2 dx dy jr Gj

˝

(6.49)

Thus, from the Poincaré inequality (6.46) with (6.44) and (6.47)–(6.49) we obtain in the end kOui k2L2 .˝/ C3 ;

(6.50)

"2 kOu3 k2L2 .˝/ C4 :

(6.51)

6.4 Limit Variational Inequality, Strong Convergence, and the Limit Equation

125

Let

@v 2 L2 .˝/ Vy D v 2 L .˝/ W @y 2

be the Banach space with norm kvkVy D

kvk2L2 .˝/

! 12 2 @v C : @y L2 .˝/

Theorem 6.2. There exist u?i 2 Vy for i D 1; 2, p? 2 L02 .˝/, and a subsequence " ! 0 such that uO i ! u?i "

weakly in Vy

for i D 1; 2;

@Oui ! 0 weakly in L2 .˝/ @xj

"

@Ou3 ! 0 weakly in L2 .˝/; @y

"2

@Ou3 ! 0 weakly in L2 .˝/ @xi pO ! p?

weakly in L2 .˝/;

for

i; j D 1; 2;

(6.52) (6.53) (6.54)

for

i D 1; 2; p? depends only on x;

"Ou3 ! 0 weakly in L2 .˝/:

(6.55) (6.56) (6.57)

Proof. From (6.44) and (6.50) we have (6.52), and from (6.44) and (6.52) we P @Ou get (6.53). By (6.53) and 2jD1 @xjj D @O@yu3 , (6.54) holds. From (6.51) and (6.44) we deduce (6.55). From (6.45) we obtain (6.56), and from (6.51), div uO D 0, and a suitable choice of the test function we obtain also (6.57). t u

6.4 Limit Variational Inequality, Strong Convergence, and the Limit Equation To obtain the variational inequality for limit functions u? , p? in ˝ we change variables in the inequality defining weak solutions in ˝ " and then using Theorem 6.2, pass to zero with ". First, we write (6.20) in new variables uO ; pO in ˝. In the place of the test function ' in ˝ " we put the new test function 'O in ˝, which is connected with ' in the same

126


O so that summing over K way as uO is connected with u. For ' 2 K we have 'O 2 K, with ' we sum over KO with '. O We have, after multiplying (6.20) by ", Z @Ouj @Oui @ @ " " .'Oi uO i / C " .'Oj uO j / dxdy C" @xj @xi @xj @xi ˝

2 1X 2 i;jD1

2 Z 1X @Oui @ u3 2 @O 2 @ C C" .'Oi uO i / C " .'O3 uO 3 / dxdy 2 iD1 ˝ @y @xi @y @xi Z Z @'O1 @Ou3 @ @'O2 @'O3 C2 " .'O3 uO 3 /dxdy dxdy " pO C C @y @y @x1 @x2 @y ˝ ˝ 2 Z X p OlOui .x; h.x//Œ'Oi .x; h.x// uO i .x; h.x// 1 C jrh" .x/j2 dx C !

iD1

Z C

!

O 'O sj jOu sj/dx 0: k.j

(6.58)

If we take all nonlinear terms in (6.58) to the right-hand side and take lim inf"!0 on both sides (in fact on the left-hand side the simple lim"!0 exists), we obtain the limit variational inequality, 2 X iD1

Z @u?i @.'Oi u?i / @'O1 @'O2 dxdy dxdy p? .x/ C @y @x1 @x2 ˝ @y ˝ Z @h @h ? p .x/ 'O1 .x; h.x// .x/ C 'O2 .x; h.x// .x/ dx @x1 @x2 ! Z 2 X Ol u?i .x; h.x// 'Oi .x; h.x// u?i .x; h.x// dx C 1 2

Z

iD1

Z C

!

!

kO .j'O sj ju? sj/ dx 0:

(6.59)

We shall confine ourselves to present the main points of the proof. 1. In the above inequality (6.59) the third components of u? and 'O do not occur, so that we can write 'N D .'O1 ; 'O2 /, and (6.59) holds for all 'N that are projections of 'O 2 KO on the first two components. Let O D f'N D .'O1 ; 'O2 / 2 .H 1 .˝//2 W 9'O3 2 H 1 .˝/; 'O D .'O1 ; 'O2 ; 'O3 / 2 Kg: O ˘.K/

6.4 Limit Variational Inequality, Strong Convergence, and the Limit Equation

127

The second line in (6.59) is written in terms of ', N and equals Z

@'O3 p .x/ dydx D @y ˝ ?

Z Z !

h.x/

0

@'O3 p .x/ dydx D @y ?

Z !

p? .x/'O3 .x; h.x//dx;

as 'O3 .x; 0/ D 0 and 'O3 D 0, on L . Since 'O1 n1 C 'O2 n2 C 'O3 n3 D 0, and 'O3 D

p @h @h 1 .'O1 n1 C 'O2 n2 / D 1 C jrh.x/j2 .'O1 n1 C 'O2 n2 / D 'O1 C 'O2 ; n3 @x1 @x2

(cf. (6.30)), we have Z

Z

@'O3 dydx D p? .x/ @y ˝

! @h p? .x/ 'Oi .x; h.x// .x/ dx: @xi ! iD1 2 X

2. Integrals over F . We know that, cf. (6.21), (6.37), and (6.38), l"

Z F"

u2 dS D O

1 ; "

and from (6.47) and (6.48) we get Z

jOui j2 dS C.h/

F

Z

F

jOu3 j2 dS

Z F"

1 C.h/ "2

whence from (6.49), Z O jOui j2 dS C.h/C.˝; Ol; G/ F

jui j2 dS

Z F"

for

i D 1; 2;

ju3 j2 dS;

Z and F

jOu3 j2 dS

1 O C.h/C.˝; Ol; G/: "2 (6.60)

From (6.60) we conclude that uO i ! vi? weakly in L2 .F /, and, since Z "l" ui .'i ui /dS F"

D Ol

Z !

uO i .x; h.x// Œ'Oi .x; h.x// uO i .x; h.x//

p 1 C jrh" .x/j2 dx;

we get the terms Ol

Z !

u?i .x; h.x// 'Oi .x; h.x// u?i .x; h.x// dx

on the left-hand side of (6.59), as with " ! 0.

p 1 C jrh" .x/j2 ! 1 strongly in L2 .!/

128


The third component (for i D 3) is nonpositive in the limit and we can omit it. In fact, Z Z " O "l u3 .'3 u3 /dS l u3 '3 dS F"

D Ol D Ol

Z Z

F"

!

!

p u3 .x; h" .x//'3 .x; h" .x// 1 C jrh" .x/j2 dx p "Ou3 .x; h.x//"'O3 .x; h.x// 1 C jrh" .x/j2 dx;

and from (6.60) f"Ou3 g is bounded in L2 .F / and p "'O3 .x; h.x// 1 C jrh" .x/j2 ! 0 in

L2 .!/;

so that lim Ol

Z

"!0

F"

u3 .'3 u3 / dS 0:

6.5 Remarks on Function Spaces In this subsection we give precise characterization (e.g., in terms of density theorems) of the admissible test functions in the limit inequality (6.59), crucial for further considerations (strong convergence of the velocity field, limit equations, uniqueness). Consider the limit variational inequality (6.59). The function space to which the test functions belong is O WD f'N D .'O1 ; 'O2 / 2 .H 1 .˝//2 W 9'O3 2 H 1 .˝/; 'O D .'O1 ; 'O2 ; 'O3 / 2 K; O ˘.K/ that is; 'O 2 .H 1 .˝//3 ; 'O GO D 0 on L ; 'O n D 0 on F [ !g: O div GO D @GO 1 C @GO 2 C @GO 3 D 0 in ˝. Denote G N D .GO 1 ; GO 2 / for Here, GO 2 K, @x1 @x2 @y O O O O in terms of O O G D .G1 ; G2 ; G3 / 2 K: We would like to characterize the space ˘.K/ O 'O1 ; 'O2 only. Instead of working directly with ˘.K/ it will be easier to consider the O WD f'N 2 .H 1 .˝//2 W 'N C G N 2 ˘.K/g: O Thus, space ˘0 .K/ O D f 'N D .'O1 ; 'O2 / 2 .H 1 .˝//2 W ˘0 .K/ 'O D .'O1 ; 'O2 ; 'O3 / D 0

on

there exists L ;

'O3 2 H 1 .˝/

'O n D 0

on

such that

F [ !g:

(6.61)

6.5 Remarks on Function Spaces

129

O in terms of .'O1 ; 'O2 /. Let 'N D .'O1 ; 'O2 / be an arbitrary We shall characterize ˘0 .K/ element of .H 1 .˝//2 such that 'O1 D 'O2 D 0 on L . We shall prove that one can always find 'O3 2 H 1 .˝/ such that .'O1 ; 'O2 ; 'O3 / D 'O satisfies condition in (6.61), that O D 0 .K/, O where is, we shall prove that ˘0 .K/ O D f'N 2 .H 1 .˝//2 W 0 .K/

'O1 D 'O2 D 0

on L g:

O then 'N 2 0 .K/, O that is, ˘0 .K/ O 0 .K/: O We shall We can see that if 'N 2 ˘0 .K/ O ˘0 .K/. O Let 'N 2 0 .K/. O We have to define 'N3 2 H 1 .˝/ such prove that 0 .K/ that 'O D .'O1 ; 'O2 ; 'O3 / D 0 on L , 'O n D 0 on F [ !. The first condition is just 'O3 D 0 on L , the second is 'O3 D 0 on ! and 'O1 .x; h.x//

@h @h .x/ C 'O2 .x; h.x// .x/ D 'O3 .x; h.x// @x1 @x2 2 H 1 .˝/ such that

Thus, we have to show existence of 'O3 D D 'O1 .x; h.x//

on

@h @h .x/ C 'O2 .x; h.x// .x/ D @x1 @x2

F :

D 0 on L [ !, on F :

1

1

As 'N D .'O1 ; 'O2 / 2 .H 1 .˝//2 , 1 is well defined on F and belongs to H 2 .F / because h 2 C2 .!/. Since 'O1 D 'O2 D 0 on L , we see that 1 D 0 on @˝. Thus, there is no singularity at @!, and the function b

D

0

on on

L [ !; F ;

1

belongs to H 2 .@˝/. Then, to prove the existence of 2 H 1 .˝/ such that on @˝ we find as the solution of the Laplace problem

D0

in ˝;

D

b

D

b

on @˝:

O D ˘0 .K/, O and we characterized ˘0 .K/ O in terms of the This proves that 0 .K/ O D functions 'O1 and 'O2 only (and not 'O3 —the third component). Since ˘.K/ O C G, N we have ˘0 .K/ N on O D f'N D .'O1 ; 'O2 / 2 .H 1 .˝//2 W 'N D G ˘.K/

L g:

O be such that the following condition, which Lemma 6.8. Let 'N D .'O1 ; 'O2 / 2 ˘.K/ will be denoted as condition (D’) holds Z @

@

'O1 .x; y/ .x/ C 'O2 .x; y/ .x/ dxdy D 0 for all @x1 @x2 ˝

2 C01 .!/:

(6.62)

130

Then


Z

@'O1 @'O2 @'O3 p .x/ C C @x1 @x2 @y ˝ ?

dxdy D 0;

which is equivalent to Z @'O1 @'O2 dxdy p? .x/ C @x1 @x2 ˝ Z @h @h ? C p .x/ 'O1 .x; h.x// dx D 0: C 'O2 .x; h.x// @x1 @x2 !

(6.63)

(6.64)

Proof. Let m 2 C01 .!/, m ! p? in L2 .!/. Then Z @'O1 @'O2 @'O3 dxdy

m .x/ C C @x1 @x2 @y ˝ ! Z Z 3 X @ m @ m D 'O1 dxdy C C 'O2

m .x/ 'Oi ni dS; @x1 @x2 ˝ @˝ iD1 as @ @ym D 0: Now, the surface integral equals zero since m 2 C01 .!/ and 'O n D 0 on F [ !. The first integral on the right-hand side equals zero by condition (D’). Thus, the integral on the left-hand side equals zero. We pass to the limit m ! p? in L2 .!/ to get (6.63). As Z Z Z h.x/ @'O3 @'O3 ? ? .x; y/ dxdy D .x; y/ dy dx p .x/ p .x/ @y @y 0 ˝ ! ! Z Z 2 X @h ? ? D p .x/'O3 .x; h.x//dx D p .x/ 'Oi .x; h.x// dx; @xi ! ! iD1 we have the equivalence of (6.63) and (6.64).

t u

The above lemma tells us that if in the limit variational inequality (6.59) we choose a test function belonging to the set O FKO WD f'N 2 ˘.K/

and

'N

satisfies condition (D’)g;

(6.65)

then this inequality will reduce to the following one, Z Z 2 X @u?i @.'Oi u?i / 1 dxdy C Ol u?i .x; h.x//.'Oi u?i /.x; h.x// dx C 2 ˝ @y @y ! iD1 Z kO .j'N sj ju? sj/ dx 0 for all 'N 2 FKO ; (6.66) !

in which the pressure is not present.


131

Let F WD fu? W u? is a weak limit of uO in the topology of Vy Vy g: O in the topology of Vy Vy . We prove that F is contained in the closure of ˘.K/ This will allow us later to get useful conclusions from variational inequalities (6.59) and (6.66), by taking test functions from F . Lemma 6.9. Each solution u? of the limit variational inequality satisfies condition .D0 /, and u? GO satisfies the following stronger condition (D) Z @

@

? ? O O .u1 G1 / .x/ C .u2 G2 / .x/ dxdy D 0 @x1 @x2 ˝

for all

2 C1 .!/: (6.67)

Proof. From div uO D 0 in ˝ we have, for 2 C01 .!/,

Z 0D

˝

.x/

@Ou1 @Ou2 @Ou3 C C @x1 @x2 @y

Z dxdy D ˝

uO 1

@

@

C uO 2 @x1 @x2

dxdy

(as uO n D 0 on F [ !). As uO i ! u?i weakly in Vy , i D 1; 2, we obtain condition (D’) for u? , that is, Z ˝

u?1

@

@

C u?2 @x1 @x2

dxdy D 0

for all

2 C01 .!/:

O n D 0 on F [ ! Taking uO GO instead of uO at the beginning of this proof, as .Ou G/ O t u and uO G D 0 on L , we obtain (6.67) in the limit. From Lemmas 6.8 and 6.9 it follows that condition (D’) is the right one to get rid of the pressure terms in the limit variational inequality. Note that in condition (D’) we do not assume that 'N 2 .H 1 .˝//2 , that is, (6.62) has sense for 'N 2 .L2 .˝//2 , for example. The solution u? satisfies (D’) and is not in .H 1 .˝//2 . We shall characterize the set F of solutions u? . n @u?i 2 L2 .˝/; i D 1; 2; F D u? D .u?1 ; u?2 / 2 .L2 .˝//2 W @y

u? satisfies .D0 /;

o u? satisfies condition .CF / coming from uO n D 0 on F :

(6.68)

Our main problem now is to find relations between FKO given by (6.65) and F given by (6.68). We do not know what is the condition .CF / in (6.68) in the general case (note that if F is given by y D const: we have no condition on u? coming from uO n D 0 on F ). Thus we define

132


?

F1 D u D

.u?1 ; u?2 /

2

2 .L .˝//

2

@u?i 2 ? 0 2 L .˝/; i D 1; 2; u satisfies .D / : @y (6.69)

Then F F1 and it is sufficient to prove the following lemma. O in the topology of Vy Vy . Lemma 6.10. F1 is contained in the closure of ˘.K/ Proof. Let v 2 F1o D fv D u? GO W u? 2 F1 g. We consider the map u D ! .0; 1/ ! ˝ W

.x; y/ 7! .x; h.x/y/ D .X; Y/;

and define v.X; Y/ D v .x; h.x/y/ D v.x; N y/

U.x; y/ D v.x; N y/h.x/:

and

Then • v and U are of the same regularity, • “v satisfies (D’) or (D) in ˝” is equivalent to “U satisfies (D’) or (D) in u.” Indeed for all 2 C1 .!/ we have Z

Z ˝

v.X; Y/r .X/dXdY D

U.x; y/r .x/dxdy; u

• “v D 0 on L ” is equivalent to “U D 0 on L ,” and also • If Un .x; y/ ! U.x; y/ in Vy .u/ Vy .u/, then vn .X; Y/ ! v.X; Y/ in Vy .˝/ Vy .˝/. Indeed, Z ˝

jvn .X; Y/ v.X; Y/j2 dxdy D

Z Z

u

D Z

u

D u

jvN n .x; y/ v.x; N y/j2 h.x/ dxdy jvN n .x; y/h.x/ v.x; N y/h.x/j2 jUn .x; y/ U.x; y/j2

1 hmin

Z u

1 dxdy h.x/

1 dxdy h.x/

jUn .x; y/ U.x; y/j2 dxdy;


133

and ˇ2 Z ˇ ˇ @vn ˇ @v ˇ ˇ dXdY .X; Y/ .X; Y/ ˇ @Y ˇ @Y ˝ ˇ Z ˇ ˇ @vN n 1 @vN 1 ˇˇ2 ˇ D .x; y/ .x; y/ h.x/ dxdy ˇ h.x/ @y h.x/ ˇ u @y ˇ2 Z ˇ ˇ 1 ˇ @vN n @vN ˇ ˇ D ˇh.x/ @y .x; y/ h.x/ @y .x; y/ˇ h3 .x/ dxdy u ˇ2 Z ˇ ˇ ˇ @Un 1 @U ˇ 3 .x; y/ .x; y/ˇˇ dxdy: ˇ @y hmin u @y Let U .x; y/ D U. x; y/, > 1, Z u

Then as .z/ D

1

2

U.x; y/r .x/ dxdy D 0 z

for all

2 C1 .!/, we have Z

Z u

2 C1 .!/:

U .x; y/r .x/dxdy D

u

Z U. x; y/r .x/dxdy D

z dzdy U.z; y/r

D 0: 2 u

Thus, U satisfies condition (D). Since it is not smooth enough in x, we take the O for usual mollification " ? U with respect to x variable only which is in ˘0 .K/ small ", " ". /. We check condition (D). Z

Z Z u

." ? U /.x; y/r .x/dydx D

!

0

Z Z D !

Z D

u

1

0

Z !

1

U .t; y/" .t x/dt r .x/dydx

U .t; y/

Z !

" .x t/r .x/dx dydt

U .t; y/r." ? /.t/dydt D 0:

As " ? 2 C1 .!/ and U satisfies condition (D), then " ?U also satisfies condition .D/ in u. Now, kU " ? U kVy2 kU U kVy2 C kU " ? U kVy2 ; and we take close to 1 first and then small ". O such that kU " ? U kV 2 .u/ ! 0, which is Thus there exists " ? U 2 ˘0 .K/ y O with kv " ? v kV 2 .˝/ ! 0. equivalent to the existence of " ? v 2 ˘0 .K/ t u y

134


6.6 Strong Convergence of Velocities and the Limit Equation From the Korn inequality (6.24) written in new variables we conclude that if uO and 'O in KO are divergence free then, in particular 2 X @.Oui 'Oi / 2 2 @y

Z

L .˝/

iD1

aO .Ou '; O uO '/ O C C.F ; h/

F

jOu 'j O 2 dS:

By (6.20), aO .Ou '; O uO '/ O D aO .'; O 'O uO / C aO .Ou; uO '/ O Z Z O O aO .'; O 'O uO / C l uO .'O uO / dS C k .j'O sj jOu sj/ dx; F

!

and we have 2 X @.Oui 'Oi / 2 2 @y

L .˝/

iD1

aO .'; O 'O uO / C Ol C kO

Z !

Z F

uO .'O uO / dS

.j'O sj jOu sj/ dx Z

C C.F ; h/

F

jOu 'j O 2 dS;

where aO .'; O 'O uO / D

2 Z @'Oj 1X @'Oi @ @ " " .'Oi uO i / C " .'Oj uO j / dxdy C" 2 i;jD1 ˝ @xj @xi @xj @xi

2 Z 1X @'Oi @ @'O3 @ C "2 .'Oi uO i / C "2 .'O3 uO 3 / dxdy 2 iD1 ˝ @y @xi @y @xi Z @'O3 @ " .'O3 uO 3 /dxdy: C2 " @y @y ˝ C

Thus 2 X @.Oui 'Oi / 2 2 @y iD1

L .˝/

C kO

Z !

.jOu sj j'O sj/dx

aO .'; O 'O uO / C Ol

Z

Z F

uO .'O uO / dS C C.F ; h/

F

jOu 'j O 2 dS:

6.6 Strong Convergence of Velocities and the Limit Equation

135

O Then Let uN D .Ou1 ; uO 2 /, u? D .u?1 ; u?2 / as in (6.52) and let 'N D .'O1 ; 'O2 / 2 ˘.K/. (

2 X @.Oui 'Oi / 2 lim sup 2 @y "!0

Z ˝

)

Z !

L .˝/

iD1

C kO

@'N @.'N u? / dxdy C Ol @y @y

Z F

.jNu sj j'N sj/dx

u? .'N u? / dS C C.F ; h/

Z F

ju? 'j N 2 dS;

and 2 X @.Oui 'Oi / 2 2 @y

L .˝/

iD1

Z

C kO Z

!

.jNu sj j'N sj/dx

Z @'N @.'N u? / dxdy C Ol u? .'N u? / dS @y ˝ @y F Z C C.F ; h/ ju? 'j N 2 dS C ı for all ı < ".ı/:

F

O which has u? as a By Lemma 6.10, there exists a sequence of functions 'N in ˘.K/ 2 limit in Vy , whence 2 X @.Oui 'Oi / 2 2 @y iD1

C kO

Z !

L .˝/

.jNu sj ju? sj/dx ı

for all

ı < ".ı/:

SinceRı > 0 is arbitrary, then by the weak lower semicontinuity of the functional v 7! ! jv sj dx, we get Z !

.jNu sj ju? sj/dx ! 0:

and from Lemma 6.5 the strong convergence of uN " to u? in VQ y2 \ F1 follows. Now, we obtain from the limit inequality the classical relations for u? and p? . Theorem 6.3. The limit functions u? ; p? satisfy p? .x1 ; x2 ; y/ D p? .x1 ; x2 /

a:e: in ˝;

1 @2 u?i @p? C D 0 .i D 1; 2/ 2 2 @y @xi

p? 2 H 1 .!/; in

L2 .˝/:

(6.70) (6.71)

136


Proof. Using Lemma 6.10 we first take 'O1 D u?1 ˙ 1 with 1 2 H01 .˝/ and 'O2 D u?2 , and then 'O1 D u?1 and 'O2 D u?2 ˙ 2 with 2 2 H01 .˝/ in (6.59), to get

1 @2 u?i @p? C D 0 in 2 @y2 @xi

H 1 .˝/;

i D 1; 2:

(6.72)

Multiplying (6.72) by i .x; y/ D y.y h.x// .x/ with 2 H01 .!/ and using the Green formula we have Z Z @u?i @ i @u?i @Œy2 yh.x/ .x/ dxdy D dxdy @y ˝ @y @y ˝ @y Z @u?i D Œ2y h.x/ .x/dydx ˝ @y Z Z ? 2ui .x; y/ .x/dydx C u?i Œ2y h.x/ .x/n3 dS D Z

˝

2h.x/Qu?i .x/ .x/dx C

D !

Z C !

Z

@˝

!

h.x/u?i .x; 0/ .x/dx

h.x/u?i .x; h.x// .x/dx;

where h.x/Qu?i .x/

Z

h.x/

D 0

u?i .x; y/dy;

and Z

@ i p .x/ dxdy D @xi ˝ ?

Z

Z

?

!

Z D !

D

p .x/ p? .x/

1 6

Z !

h.x/ 0

@ @xi

p? .x/

! @Œy2 yh.x/ .x/ dy dx @xi ! Z

.x/

h.x/

0

Œy2 yh.x/dy dx

@h3 .x/ .x/ dx: @xi

Thus h.x/Qu?i .x/ C

h.x/ ? h.x/ ? h3 .x/ @p? ui .x; h.x// C ui .x; 0/ D .x/ 2 2 6 @xi

(6.73)

in H 1 .!/. As u?i 2 Vy , uQ ?i 2 L2 .!/ and in consequence the left side of (6.73) belongs to L2 .!/. This gives (6.70) and (6.71). t u

6.7 Reynolds Equation and the Limit Boundary Conditions

137

6.7 Reynolds Equation and the Limit Boundary Conditions Theorem 6.4. The pair .u? ; p? / satisfies the following weak form of the Reynolds equation Z !

Z Z h3 ? h ? h ? ? rp s sh r'dx C sh rh'dx D ' gQ n dl 12 2 2 ! @!

(6.74)

for all ' 2 H 1 .!/, where s? .x/ D u? .x; 0/;

s?h .x/ D u? .x; h.x//;

and Z gQ .x/ D

h.x/ 0

gO .x; y/dy

for all

?

x 2 @!:

?

Moreover, the traces ? D @u@y .; 0/, h? D @u@y .; h.x//, s?h D u? .; h.x//, and s? D u? .; 0/ satisfy the following limit form of the Tresca boundary conditions (6.4) and the Fourier boundary condition (6.6) j ? j D kO H) 9 0 u? D s C ? o a:e: in !; j ? j < kO H) u? D s h? rh n C Ols?h D 0

a:e: on

F :

(6.75) (6.76)

Proof. Integrating twice (6.71) between 0 and y we obtain (for 1 i 2) u?i .x; y/ D

y2 @p? .x/ C s?i .x/ C yi? .x/; 2 @xi

(6.77)

and for y D h.x/ we get s?h .x/ D

h2 ? rp .x/ C s? .x/ C h ? .x/ 2

a:e: in !:

Now, integrating twice (6.71) between y and h.x/ we obtain (for 1 i 2) u?i .x; y/ D

y2 h2 hy C 2 2

@p? .x/ ? C s?h;i .x/ C .y h/h;i .x/; @xi

(6.78)

138


and for y D 0 we get s? .x/ D

h2 ? rp .x/ C s?h .x/ hh? .x/ 2

a:e: in !:

(6.79)

Taking now the average Z

1 v.x/ Q D h.x/

h.x/ 0

v.x; y/ dy

of (6.77) we obtain hQu?i .x/ D

h3 @p? .x/ h2 C hs?i .x/ C i? .x/: 6 @xi 2

Moreover, for all ' 2 H 1 .!/, we have Z

Z 0D

˝

' div uO dydx D

!

Z '.x/

Z D !

'.x/

2 X @Oui

h 0

iD1

@Ou3 C @xi @y

2 X @.huQO i /

@xi

iD1

! dydx !

C uO 3 .x; h/ uO 3 .x; 0/ dx:

Then as uO n D uO 3 .x; 0/ D 0 on !, and F is defined by y D h.x/, we obtain uO 3 .x; h.x// D uO 1 .x; h.x//

@h @h C uO 2 .x; h.x// ; @x1 @x2

and Z !

'.x/

2 X @.huQO i / iD1

@xi

@h C uO i .x; h.x// @xi

! dx D 0:

Using the Green formula we have

2 Z X iD1

@' huQO i dx C @xi !

Z

2 X

@h '.x/ uO i .x; h.x// @xi ! iD1

! dx C

2 Z X iD1

@!

huQO i ni ' dl D 0:

As uO i ! u?i weakly in Vy , uQO i ! uQ ?i weakly in L2 .!/, and as @! @˝, we obtain Z

huQO r'dx !

Z !

Z '.x/ .Ou.x; h.x//rh/ dx D

@!

gQ .x/ n' dl

for all

' 2 H 1 .!/:

6.7 Reynolds Equation and the Limit Boundary Conditions

139

From (6.79) we have Z Z Z 3 h h2 ? ? ? ? ' gQ n dl rp C hs C r' dx 'sh rh dx D 6 2 ! ! @!

(6.80)

for all ' 2 H 1 .!/. Then using (6.78) and (6.80), we obtain the weak formulation (6.74) of the Reynolds equation. To prove (6.75) we transform (6.59). Using the Green formula, (6.71), conditions 'i D gO i on L , ni D 0 on ! for i D 1; 2, and ' n D 0 on ! [ F , as well as the O and of D.˝/ in L2 .˝/, we obtain density of D.˝/ in ˘.K/, Z O k.j'.x; 0/ sj js? sj/ dx !

Z C Z

!

C !

Ols?h .'.x; h.x// s?h / dx

Z !

? .'.x; 0/ s? / dx

h? .'.x; h.x// s?h /rh n dx 0 for all

' 2 .L2 .˝//2 :

(6.81)

We take first '.x; y/ D

s?h .x/ si .x/

if y D h.x/; if y D 0;

then '.x; y/ D

s?h .x/ 2s?i si

if y D h.x/; if y D 0;

in (6.81), to obtain Z O ? sj ? .s? s/ dx D 0; kjs

(6.82)

!

and from (6.81), with '.x; y/ D

s?h .x/ .x/ C s

if if

y D h.x/; y D 0;

and 2 .L2 .!//2 , we obtain Z Z Okjj ? dx O ? sj ? .s? s/ dx: kjs !

(6.83)

!

From (6.82) and (6.83) we deduce Z O ? dx 0 kjj !

for all 2 .L2 .!//2 :

(6.84)

140


We will show that j j kO a.e. on !. Suppose that there exists a set of positive measure !1 ! such that j j > kO on this set. Taking D 0 on ! n !1 and D on ! in (6.84) we get Z !1

j j.kO j j/ dx 0:

As the integrand must be a strictly negative function, we arrive at a contradiction. We have thus proved j ? j kO

a:e: on

!:

Then O ? sj j ? jjs? sj ? .s? s/ a:e: on kjs

!

and O ? sj ? .s? s/ 0 a:e: on kjs

!:

From (6.82) we deduce that O ? sj ? .s? s/ D 0 kjs

a:e: on

!:

(6.85)

O then from (6.85) we have j ? jjs? sj D ? .s? s/ a.e. on !, which If j ? j D k, implies the existence of 0 such that s? s D ? . O then from (6.85) we have If, in turn, j ? j < k, O ? sj ? .s? s/ .kO j ? j/js? sj a:e: on 0 D kjs

! \ fj j < kg;

whence s? s D 0 a.e. on ! \ fj j < kg, and (6.75) follows. We shall prove (6.76). First we take '.x; y/ D

.x/ s? .x/

if y D h.x/ if y D 0

with 2 .L2 .!//2 in (6.81), to get Z Ols?h . s?h / C h? . s?h /rh n dx 0 for all !

Then, taking D

C s?h and D

C s?h we obtain

Z Ols?h C h? rh n dx D 0 for all !

which implies (6.76).

2 .L2 .!//2 :

2 .L2 .!//2 ; t u

6.8 Uniqueness

141

Remark 6.5. From formula (6.74) we obtain the following form of the Reynolds equation (for D 1), div

h3 ? h ? h ? rp s sh C s?h rh D 0 12 2 2

in H 1 .!/:

In the simplest classical one-dimensional case, with ! D .a; b/, s?h D 0, s? D const, we obtain the well-known equation describing the pressure distribution in the interval .a; b/,

d dx

h3 dp? 6 dx

C s?

dh D 0: dx

(6.86)

Observe how the Reynolds equation depends on the boundary conditions for the velocity field. When 6D 1, the second term on the left-hand side of (6.86) should be multiplied by .

6.8 Uniqueness Theorem 6.5. There exists a unique solution .u? ; p? / in VQ y .L02 .!/ \ H 1 .!// of inequality (6.59). Proof. Let .U 1 ; p1 / and .U 2 ; p2 / be two solutions of (6.59). Taking ' D U 2 and ' D U 1 , respectively, as test functions in (6.59) we reduce it to (6.66) and obtain 1 2

ˇ Z Z ˇ ˇ @.U 1 U 2 / ˇ2 ˇ dxdy C Ol jU 1 .x; h.x// U 2 .x; h.x//j2 dx 0: ˇ ˇ ˇ @y ˝ !

Using Lemma 6.5 we get kU 2 U 1 kVy2 D 0

and

jU 1 .x; h.x// U 2 .x; h.x//j D 0

a:e: in !:

(6.87)

Moreover, from (6.74), we have Z !

h3 r.p2 p1 /r'.x/dx 6 Z D h.x/ŒU 1 .x; 0/ U 2 .x; 0/ C U 1 .x; h.x// U 2 .x; h.x//r'.x/dx !

Z

C !

h.x/.U 2 .x; h.x// U 1 .x; h.x///rh 'dx

for all

' 2 H 1 .!/:

142


Now, taking ' D p2 p1 , and by (6.87) we get kr.p2 p1 /kL2 .!/ D 0; and from the Poincaré inequality, as pi 2 L02 .!/ we conclude that kp2 p1 kL2 .!/ D 0. u t The uniqueness of u? follows then from (6.59).

6.9 Comments and Bibliographical Notes (1) Remarks on boundary conditions. The right choice of boundary conditions for the velocity field in hydrodynamical problems is fundamental for applications. In lubrication problems the widely assumed non-slip condition when the fluid has the same velocity as surrounding solid boundary is not respected any more when the shear rate becomes too high 107 s1 . This phenomenon has been studied in a lot of mechanical papers, e.g., [120, 121, 214, 223]. Continuous experimental studies are conducted (e.g., [193]) but are still difficult due to thickness of the gap between the solid surfaces which can be as small as 50 nm. In such operating conditions, non-slip condition is induced by chemical bounds between the lubricant and the surrounding surfaces and by the action of the normal stresses, which are linked to the pressure inside the flow. On the contrary, tangential stresses are so high that they tend to destroy the chemical bounds and induce slip phenomenon. This is nothing else than a transposition of the well-known Coulomb law between two solids [90] to the fluid–solid interface. Although being implicitly used in numerical procedures for lubrication problem, a Reynolds thin film equation taking account of such slip phenomenon have been hardly studied from the mathematical point of view. In [15, 21], respectively, the simplified Tresca and Coulomb interface conditions, posed on only one of the surrounding surfaces of a Newtonian fluid, were considered. This chapter is based on the results from [23]. Corresponding results for non-Newtonian fluids have been obtained in [20]. (2) Comments on technical problems. Two technical issues were crucial in the mathematical analysis of the problem. First, we could not use the usual Korn inequality as we did not assume that the velocity was equal to zero at one of the boundaries (on the top or the bottom) as is usually assumed in lubrication problems. We thus derived an analogue of the Korn inequality suitable for our boundary conditions and such that the constants could be controlled appropriately as the gap between the surfaces approached zero. This led us to the main uniform estimate of the velocity fields and to the limit variational inequality, in consequence. Second, to be able to make use of the latter, we had to characterize precisely the limit solution space and the set of admissible test functions. As the limit variational inequality was written in terms of the first two components of the velocity field, we had to characterize—in this very limit case—projections of the convex sets appearing in the weak form of the Stokes flow. This allowed us, in particular, to obtain a stronger convergence of the velocity fields than it is usually expected.

7

Autonomous Two-Dimensional Navier–Stokes Equations

We put our faith in the tendency for dynamical systems with a large number of degrees of freedom, and with coupling between those degrees of freedom, to approach a statistical state which is independent (partially, if not wholly) of the initial conditions. – George Keith Batchelor

This chapter contains some basic facts about solutions of nonstationary Navier–Stokes equations @u 1 C .u r/u D rp C u C f ; @t div u D 0; in two-dimensional spatial domains. We assume that the system is autonomous, that is, the external forces do not depend on time and that the domain of the flow is bounded. The boundary conditions are either periodic or homogeneous of Dirichlet type. In what follows, the viscosity is a positive constant and we set D 1. First, we prove the existence of a unique global in time weak solution. Then we analyze the time asymptotics of solutions proving the existence of the global attractor and showing its trivial structure in some cases of special external forces. Finally, we consider the problem of behavior of the kinetic energy of the fluid. To this end we introduce the notion of different scales of the flow, considered in the theory of turbulence, and corresponding in a way to different spatial dimensions of vortices of a given turbulent flow.

7.1 Navier–Stokes Equations with Periodic Boundary Conditions Let Q D Œ0; L2 and define spaces H D fu D .u1 ; u2 / 2 LP p2 .Q/ W div u D 0g; © Springer International Publishing Switzerland 2016 G. Łukaszewicz, P. Kalita, Navier–Stokes Equations, Advances in Mechanics and Mathematics 34, DOI 10.1007/978-3-319-27760-8_7

143

144

7 Autonomous Two-Dimensional Navier–Stokes Equations

and P p1 .Q/ W div u D 0g; V D fu D .u1 ; u2 / 2 H (cf. Sect. 3.4) with the scalar products Z

Z u.x/ v.x/ dx

.u; v/ D

in H;

..u; v// D

Q

ru.x/ W rv.x/ dx

in V;

Q

and the norms sZ ju.x/j2 dx;

kukH D juj D

juj2 D .u; u/;

Q

and sZ jru.x/j2 dx;

kukV D kuk D

kuk2 D ..u; u// D .ru; ru/;

Q

respectively. The duality between V and its dual V 0 is denoted by h; i. The weak formulation of the initial boundary value problem for the Navier– Stokes equations is given in Theorem 7.1 below. We adopt the notation from Chap. 4. In particular, the Stokes operator A and the trilinear form b as well as the associated nonlinear operator B are defined in this chapter the same as in Chap. 4 (with the obvious difference that in Chap. 4 the stationary problem is threedimensional and in present chapter we consider the two-dimensional problem). Our aim is to prove the following existence and uniqueness result. P Then there exists a unique function u W Œ0; 1/ ! H Theorem 7.1. Let u0 ; f 2 H. such that u 2 C.Œ0; TI H/ \ L2 .0; TI V/;

u.0/ D u0 ;

for all T > 0, satisfying the following identity d .u.t/; v/ C ..u.t/; v// C b.u.t/; u.t/; v/ D .f ; v/ dt

for all

v 2 V;

(7.1)

in the sense of scalar distributions on .0; 1/. Moreover, the function u W Œ0; 1/ ! H is bounded and the map u0 ! u.t/ is continuous as a map in H. P1 O O Proof. Let f .x/ D jD1 fj !j .x/, fj D .f ; !j /, where !j , j D 1; 2; : : :, is an orthonormal sequence in H, a sequence of eigenvectors of the Stokes operator. We seek our solution u as a suitable limit of the sequence of approximate solutions un , n D 1; 2; : : :, which are finite dimensional solutions of the following problems.

7.1 Navier–Stokes Equations with Periodic Boundary Conditions

145

P P Problem 7.1. Let n 2 N, fn .x/ D njD1 fOj !j .x/, un .0/ D u0n D njD1 .u0 ; !j /!j .x/. On the interval .0; 1/ find un .t/ D un .; t/ of the form un .x; t/ D

n X

unj .t/!j .x/

jD1

such that d .un .t/; !j / C ..un .t/; !j // C b.un .t/; un .t/; !j / D .fn ; !j / dt un .0/ D u0n :

for j D 1; : : : ; n; (7.2)

Observe that (7.2) is a system of n ordinary differential equations for the vector functions t ! .un1 .t/; : : : ; unn .t// with initial condition ..u0 ; !1 /; : : : ; .u0 ; !n //. From the classical theorem about existence and uniqueness of solutions of the system xP D F.x/;

x.0/ D x0 ;

with polynomial (hence locally Lipschitz) right-hand side it follows existence of a unique local in time solution un .t/ on some interval Œ0; tn /, tn > 0, for every positive integer n. We shall show that the solutions un are defined in fact for all positive times (are global in time). This will follow from the first energy inequality. We multiply the jth equation in (7.2) by unj .t/ and add the resulting n equations to get 1d jun .t/j2 C kun .t/k2 C b.un .t/; un .t/; un .t// D .fn ; un .t//: 2 dt As b.un .t/; un .t/; un .t// D 0, using the Cauchy inequality to the right-hand side as well as the Poincaré inequality, cf. (3.16), juj Ckuk; we obtain the first energy inequality d jun .t/j2 C kun .t/k2 C./jfn j2 ; dt

(7.3)

d jun .t/j2 C k1 ./jun .t/j2 C./jfn j2 : dt

(7.4)

and

146


Integrating (7.3) in an interval .0; t/ we obtain jun .t/j2 C

Z

t 0

kun .t/k2 ds C./tjfn j2 C jun .0/j2 C./tjf j2 C ju.0/j2 :

Hence, for each t > 0, jun .t/j < 1 and thus the local solution can be extended to the whole positive semiaxis. From this inequality we can see that for each T > 0 our approximate solutions are uniformly bounded in the space C.Œ0; TI H/ \ L2 .0; TI V/: Moreover, the approximate solutions are infinitely differentiable functions in x and in t. Exercise 7.1. Prove that if f D 0 then any approximate solution, that is, solution starting from an arbitrary initial data u0 2 H, converges at least exponentially fast to zero in H. To get the boundedness in H of our approximate solutions we shall use the Gronwall inequality (cf. Lemma 3.20). In our case, from the Gronwall inequality applied to (7.4), we have 2

2 k1 t

jun .t/j jun .0/j e

Z

t

C 0

ek1 .st/ C./jfn j2 dt

D jun .0/j2 ek1 t C

C./ 2 jfn j .1 ek1 t /: k1

Therefore, every approximate solution is bounded on Œ0; 1/ as a function with values in H. From the uniform boundedness of the sequence of approximate solutions in C.Œ0; TI H/ \ L2 .0; TI V/ it follows that there exists a subsequence (denoted again by .un /) and some u in L1 .0; TI H/ \ L2 .0; TI V/ such that un ! u

weakly in L1 .0; TI H/;

(7.5)

which means that Z

T

Z

Z un .x; t/'.x; t/dxdt D

lim

t!1 0

Q

T

Z u.x; t/'.x; t/dxdt

0

Q

for all ' from L1 .0; TI H/, and un ! u weakly in L2 .0; TI V/;

(7.6)


147

which means that Z

Z

Z

T

un .x; t/'.x; t/dxdt D

lim

t!1 0

Z

T

u.x; t/'.x; t/dxdt 0

Q

Q

for all ' from L2 .0; TI V/. We shall prove that this limit u is a solution of Problem 7.1. From (7.2) we have Z

T

0

Z

.un .t/; v/ 0 .t/dt C Z C

0

0 T

T

..un .t/; v//.t/dt Z

b.un .t/; un .t/; v/.t/dt D

T 0

.fn ; v/.t/dt

for all v in Vn D spanf!1 ; : : : ; !n g and in C01 ..0; T//. We shall show that this identity is satisfied for the limit function u and for all v in V. As for the linear terms, we have, for any v 2 Vm , where m is any given integer, Z

T

lim

n!1 0

.un .t/; v/ 0 .t/dt D lim Z

T

D

T

Z

n!1 0

Z

0

Z

un .x; t/v.x/ 0 .t/dxdt Q

u.x; t/v.x/ 0 .t/dxdt D

Z

Q

T 0

.u.t/; v/ 0 .t/dt;

which follows from the definition of the weak-star convergence in L1 .0; TI H/, as v.x/ 0 .t/ 2 L1 .0; TI H/. Similarly, Z lim

n!1 0

T

Z ..un .t/; v//.t/dt D lim Z

T

D

run .x; t/rv.x/.t/dxdt

n!1 0

Z

0

Z

T

Q

Z

ru.x; t/rv.x/.t/dxdt D 0

Q

T

..u.t/; v//.t/dt;

as un ! u weakly in L2 .0; TI V/ and v.x/.t/ 2 L2 .0; TI V/. Moreover, Z lim

n!1 0

T

Z .fn ; v/.t/dt D

0

T

.f ; v/.t/dt;

as fn ! f strongly in L2 .Q/. The only problem which remains is that with the convergence of the nonlinear term. Weak convergences of un to u as in (7.5) and (7.6) are not sufficient to pass to the limit in the nonlinear term. This is a typical problem when dealing with nonlinear terms. Exercise 7.2. Prove that, in general, if un ! u and vn ! v weakly in L2 .Q/ then it does not follow that un vn ! uv in the sense of distributions on Q.

148


However, to prove that Z lim

n!1 0

T

Z b.un .t/; un .t/; v/.t/dt D

T 0

b.u.t/; u.t/; v/.t/dt

(7.7)

it suffices to know that un ! u strongly in L2 .Q/

(7.8)

for the above subsequence of the sequence of approximate solutions (or for some of its subsequences). Lemma 7.1. Let a.; / be a continuous bilinear form in the Banach space X with norm k k, un ! u strongly and vn ! v weakly in X. Then limn!1 a.un ; vn / D a.u; v/. Proof. Since the sequence un is strongly convergent and the sequence vn is bounded as weakly convergent, it is a.un ; vn / a.u; v/ D a.un u; vn / C a.u; vn v/ and ja.un u; vn /j Ckun ukkvn k ! 0. As for the second term, there exists u 2 V 0 , the dual of V such that a.u; vn v/ D hu ; vn viV 0 V ! 0 since the sequence vn is weakly convergent in H. t u Exercise 7.3. Prove (7.7) using (7.6), (7.8), and Lemma 7.1. Exercise 7.4. Prove (7.8) by proving first that the sequence

f@t un g is bounded in

L2 .0; TI V 0 /

and then using Theorem 3.18 and Lemma 7.1. We have thus Z 0

T

Z

.u.t/; v/ 0 .t/dt C Z C

0 T

0

T

..u.t/; v//.t/dt

(7.9) Z

b.u.t/; u.t/; v/.t/dt D

T 0

.f ; v/.t/dt

for all v in Vm D spanf!1 ; : : : ; !m g, where m is any positive integer and is in C01 .0; T/. Exercise 7.5. Prove that (7.9) holds for all v 2 V using the fact that the union of all the spaces Vm is dense in V. To prove that u 2 C.Œ0; TI H/ for all T > 0 we apply Lemma 3.16. Exercise 7.6. Prove, using the du Bois-Reymond lemma, that (7.9) implies (7.1). Exercise 7.7. Prove that u.0/ D u0 .


149

Exercise 7.8. Prove that we have the following equality in V 0 , du.t/ C Au.t/ C B.u.t// D f ; dt for a.e. t 2 .0; T/. Use Proposition 3.2 and property (3.26) of the evolution triple V H V 0. To prove the uniqueness of a solution let us assume, on the contrary, that there exist two different solutions, u and v, to our problem du.t/ C Au.t/ C B.u.t// D f ; dt

u.0/ D u0

on the interval Œ0; 1/. Setting w.t/ D u.t/ v.t/ we have dw.t/ C Aw.t/ C B.u.t// B.v.t// D 0 in V 0 : dt Thus, in particular,

dw.t/ C Aw.t/ C B.u.t// B.v.t//; w.t/ D 0; dt that is, see Theorem 3.16, 1d jw.t/j2 C kw.t/k2 C b.u.t/; u.t/; w.t// b.v.t/; v.t/; w.t// D 0: 2 dt Further, using a property of the trilinear form b, we obtain 1d jw.t/j2 C kw.t/k2 D b.w.t/; v.t/; w.t//: 2 dt We shall estimate the right-hand side applying to it the Ladyzhenskaya inequality (cf. Exercise 4.8), kukL4 .Q/2 Cjuj1=2 kuk1=2

for

u 2 V:

We have jb.u; v; w/j kukL4 .Q/2 kvkkwkL4 .Q/2 c1 juj1=2 kuk1=2 kvkjwj1=2 kwk1=2 : Therefore, b.w.t/; v.t/; w.t// c1 jw.t/kjv.t/kkw.t/k c2 kw.t/k2 C jw.t/j2 kv.t/k2 : 2

150


We have then c2 d jw.t/j2 C kw.t/k2 jw.t/j2 kv.t/k2 : dt Now, in view of the Gronwall inequality we have, for t 0, Z t c2 kv.s/k2 ds : jw.t/j2 jw.0/j2 exp 0 As w.0/ D 0, u.t/ D v.t/ for all t > 0 which proves the uniqueness. From the uniqueness result together with our construction of a weak solution on an arbitrary interval .0; T/ follows the existence of a unique global solution defined on the interval .0; 1/. Observe that from the above inequality it follows also the continuous dependence of solutions on the initial data: the map u0 ! u.t/ is continuous as a map in H for t > 0. This completes the proof of Theorem 7.1. t u Remark 7.1. One can prove that under the assumptions of Theorem 7.1 the solution u belongs to the space L2 .; TI D.A// for all T > 0 and 0 < < T, where D.A/ is the domain of the Stokes operator, cf. Sect. 4.1. Exercise 7.9. Compute the constants C./, k1 ./ in inequality (7.4). Exercise 7.10. Prove an existence and uniqueness theorem analogous to Theorem 7.1 for the case of homogeneous Dirichlet boundary conditions u D 0 at @˝ .0; T/, where ˝ is a bounded open set in R2 , and the spaces H and V in Theorem 7.1 are accordingly modified. Exercise 7.11. Consider the nonhomogeneous Dirichlet boundary condition u D a at @˝ .0; T/, where ˝ is a bounded open set in R2 with sufficiently smooth boundary, and where the function a is defined in ˝, a 2 H 1 .˝/, div a D 0, so that u a 2 V. Formulate and prove an existence and uniqueness theorem analogous to Theorem 7.1 for this case. Exercise 7.12. Non-autonomous problem. Using the argument of the proof of Theorem 7.1 prove the following result. Theorem 7.2. Let V H V 0 be either given by the homogeneous Dirichlet boundary conditions on an open and bounded set ˝ R2 with Lipschitz boundary or the periodic boundary conditions on a two-dimensional box. Let u0 2 H and 2 f 2 Lloc .RC I V 0 /. Then there exists a unique function u W RC ! H such that u 2 L2 .0; TI V/ \ C.Œ0; TI H/;

u.0/ D u0 ;

for all T 0, satisfying the identity d .u.t/; v/ C ..u.t/; v// C b.u.t/; u.t/; v/ D hf .t/; vi for all v 2 V dt

7.2 Existence of the Global Attractor: Case of Periodic Boundary Conditions

151

in the sense of scalar distributions on .0; 1/. Moreover, the map u0 ! u.t/ is continuous as a map in H. Remark 7.2. Consider either the two-dimensional periodic problem or the homogeneous Dirichlet problem on the bounded domain ˝ R2 of class C2 . Let 2 f 2 Lloc .RC I H/ and u0 2 H. Then the solution u given in Theorem 7.2 belongs 2 to L .; TI D.A// for all 0 < < T < 1 (see, for example, Theorem 7.4 in [99] or Theorem III.3.10 in [219]). Remark 7.3. For the case of homogeneous Dirichlet boundary conditions, the existence results of Theorems 7.1 and 7.2 can be extended to the case of unbounded domains, where the embedding V H is not compact. For the treatment of this case see, e.g., [201, 219].

7.2 Existence of the Global Attractor: Case of Periodic Boundary Conditions We shall present first some notions from the theory of dynamical systems. Definition 7.1. Let X be a complete metric space. A family of maps S.t/ W X ! X, t 2 Œ0; 1/, denoted as fS.t/gt0 is called a semigroup in X if S.0/ is the identity map and for all s; t > 0 it is S.t C s/ D S.t/S.s/. A semigroup is continuous in X if all the maps are continuous in X. Definition 7.2. Let fS.t/gt0 be a continuous semigroup in a complete metric space X. Then a subset A of X is a global attractor for this semigroup in X if the following conditions hold: .i/ invariance: S.t/A D A for all t > 0, .ii/ compactness: A is a nonempty and compact subset of X, .iii/ attraction: for every bounded subset B of X and every " > 0 there exists t0 D t0 .B; "/ such that for all t t0 , S.t/B O" .A /, where O" .A / is the "-neighborhood of A , O" .A / D

[

B.x; "/;

x2A

and B.x; "/ is the open ball in X with radius " and centered at x. Definition 7.3. The set A X shall be called invariant with respect to the semigroup fS.t/gt0 if S.t/A D A for all t 0. Definition 7.4. The Hausdorff semi-distance in the metric space X between two nonempty sets A; B X is defined as distX .A; B/ D sup inf .x; y/: x2A y2B

152


Exercise 7.13. Prove that the last property in the definition of the global attractor (attraction) is equivalent to say that for every bounded set B X we have lim distX .S.t/B; A / D 0:

t!1

Definition 7.5. A subset B in X is absorbing if for every bounded subset G in X there exists a time t0 D t0 .G/ such that for all t t0 , S.t/G B. Schematic illustration of a global attractor and an absorbing set is depicted in Fig. 7.1. Definition 7.6. A semigroup fS.t/gt0 in X is uniformly compact ifSfor every bounded subset G in X there exists a time t0 D t0 .G/ such that the set tt0 S.t/G is precompact in X. T S Exercise 7.14. By !.B/ D s0 ts S.t/B we define the !-limit set !.B/ of a subset B of X (cf. Fig. 7.2 for an illustration of an !-limit set). Prove that 2 !.B/ if and only if there exists a sequence n 2 B and a sequence tn ! 1 such that S.tn /n ! as n ! 1. We shall prove our main theorem about existence of a global attractor. Theorem 7.3. Let fS.t/gt0 be a continuous semigroup in a complete metric space X. If there exists a bounded absorbing set B X and the semigroup is uniformly compact in X, then the !-limit set of B, A D !.B/, is a global attractor for this semigroup in X. Proof. Let B be an absorbing subset in X.

T S Step 1. !.B/ is nonempty and compact in X. As !.B/ D s0 ts S.t/B, where S the sets ts S.t/B are nonempty and compact for s t0 .B/ (the semigroup is uniformly compact and B is bounded) and constitute a nonincreasing family

Fig. 7.1 The absorbing set B and the global attractor A for the semigroup fS.t/gt0 . All trajectories after some time (depending on the initial condition u0 ) are in the set B. They become closer and closer to the attractor A as the time goes to infinity


153

Fig. 7.2 Illustration of the !-limit set of the point u0 . The set !.fu0 g/ consists of all cluster points of all sequences fS.tn /u0 g for tn ! 1

of sets as s grows, then !.B/ is not empty and compact in view of the Cantor theorem. Step 2. !.B/ is invariant: S.t/!.B/ D !.B/ for all t > 0. ()) S.t/!.B/ !.B/. Let S.t/ D , 2 !.B/. We shall show that 2 !.B/. For there exist sequences: n in B and tn ! 1 such that S.t/ n ! . Hence, S.t/S.tn / n D S.t C tn / n ! S.t/ D (as the operator S.t/ is continuous). Moreover, 2 !.B/ by definition of the !-limit set, as the sequence S.t C tn / n converges to , where t C tn ! 1. (() !.B/ S.t/!.B/. Let 2 !.B/. We shall show that there exists 2 !.B/ such that S.t/ D . Let S.t / ! , t ! 1, 2 B. As the sets n n n n S stn t S.s/B are precompact for tn t t0 thus the sequence S.tn t/n contains a convergent subsequence S.t t/ ! for some 2 X. But 2 !.B/ as 2 B and t t ! 1. Moreover, S.t/S.t t/ D S.t / ! D S.t/ , from the continuity of S.t/. Step 3. A D !.B/ attracts bounded subsets G of X. Assume that this is not true. Then distX .S.t/G; A / ı > 0 for t ! 1, that is, there exist sequences tn ! 1 and n 2 G such that distX .S.tn /n ; A / ı > 0 for all n. On the other hand, the sequence S.tn /n contains a convergent subsequence, S.t / ! , 2 A D !.B/ as S.t / D S.t t/S.t/ and for t such that S.t/G B, S.t/ 2 B. t u Exercise 7.15. Prove that A is a maximal, in the sense of the inclusion relation, bounded invariant set. Solution. Assume that A A1 , A1 different from A , bounded and invariant. As B absorbs A1 , S.t/A1 B for large t, then A1 B as S.t/A1 D A1 from our assumption. Hence, !.A1 / D A1 !.B/ D A , and we have a contradiction.

154


Exercise 7.16. Prove that A is a connected set provided X is connected. Solution. Assume that this is not true. Then there exist open sets U1 , U2 such that Ui \ A 6D ;, i D 1; 2, A U1 [ U2 , U1 \ U2 D ;. Let B be the closed convex hull of A . Then B is compact (and so bounded) and connected. From the continuity of S.t/, the sets S.t/B are connected. The set A is a subset of each S.t/B, as A B ) A D S.t/A S.t/B. As A attracts bounded sets, distX .S.t/B; A / ! 0 as t ! 1. This means that S.t/B U1 [ U2 —an open neighborhood of A . But then we have ; 6D Ui \ A Ui \ S.t/B, i D 1; 2, S.t/B U1 [ U2 , U1 \ U2 D ;, which means that S.t/B is not connected. We have come to a contradiction. Exercise 7.17. Prove that A is a minimal closed set which attracts bounded sets of X. Solution. Assume, to the contrary, that there exists a bounded set F A , F 6D A , attracting bounded sets. As A is compact then F is also compact. Let y 2 A n F. For small ", O" .y/ \ O" .F/ D ;. Let B be an open absorbing set. As F attracts B, S.t/B O" .F/ for large t t."/. But y 2 !.B/ D A whence y D limk!1 S.tk /xk , xk 2 B, tk ! 1, and we have S.tk /xk O" .y/ for large k. But we have also S.tk /xk 2 O" .F/ for large k. A contradiction as O" .y/ \ O" .F/ D ;. Remark 7.4. The assumption of uniform compactness in Theorem 7.3 can be replaced by a weaker assumption of asymptotic compactness. However, we shall use the stronger version as in the case of the Navier–Stokes problem that we consider in this chapter the uniform compactness property holds and is easy to check. In the case of the Navier–Stokes equations in unbounded spatial domains the uniform compactness property does not hold but we can use the asymptotic compactness property [201]. Definition 7.7. We call a semigroup fS.t/gt0 asymptotically compact if it has the following property: for any bounded sequence fun g, and any sequence ftn g, n 2 N such that tn ! 1 as n ! 1, the sequence fS.tn /un g contains a convergent subsequence. Exercise 7.18. Prove that uniform compactness implies asymptotic compactness. Consider the proof of Theorem 7.3 and show that the condition of uniform compactness in the hypothesis of the theorem can be replaced by the condition of asymptotic compactness. Exercise 7.19. Prove that existence of a global attractor for a given semigroup implies existence of a bounded absorbing set and asymptotic compactness of the semigroup. The global attractor for the semigroup governed by the solution map of the ODE xP D jxj.1 x/ is presented in Fig. 7.3. Now we can come back to our main considerations. From Theorem 7.1 it follows that the family of maps S.t/ W H ! H defined for all t 0 by


155

Fig. 7.3 Example of a semigroup S.t/ W R ! R given by the solution map for the equation xP D jxj.1 x/. The global attractor is given by A D Œ0; 1. It contains two stationary points f0; 1g and all points belonging to a complete trajectory that connects them

S.t/u0 D u.t/;

(7.10)

where u./ is the unique weak solution of the considered problem with u.0/ D u0 , is a continuous semigroup in X. Our aim is to prove the following Theorem 7.4 (Existence of a Global Attractor in H). Under the assumptions of Theorem 7.1 there exists a global attractor A for the semigroup fS.t/gt0 in H associated with the two-dimensional Navier–Stokes equations with periodic boundary conditions. Proof. Step 1. Existence of a bounded absorbing set in H. From the first energy inequality, cf. inequality (7.3), d ju.t/j2 C ku.t/k2 C./jf j2 ; dt

(7.11)

we have as an immediate consequence, cf. inequality (7.4), d ju.t/j2 C k1 ./ju.t/j2 C./jf j2 : dt Applying the Gronwall inequality to the last one we have, Z t C./ 2 2 2 k1 t ju.t/j ju0 j e C ek1 .st/ C./jf j2 dt D ju0 j2 ek1 t C jf j .1 ek1 t /; k1 0 (7.12) from which it follows that for 2 > 02 D C./ jf j2 , balls B.0; / in the phase space k1 H absorb bounded sets in H. Let u0 2 B1 be a bounded subset in H. Then, ju.t/j2 2 for t t0 .B1 /. Exercise 7.20. Let u0 2 B.0; r/. Compute t0 D t0 .B.0; r// such that ju.t/j2 2 for t t0 . Step 2. Existence of a bounded absorbing set in V and uniform compactness of the semigroup. To prove these properties we need the second energy inequality. Taking

156


the scalar product of the Navier–Stokes equation with Au (for u 2 D.A/; Au D u, cf. Remark 7.1) we obtain d kuk2 C 2jAuj2 D 2.f ; Au/; dt

(7.13)

as b.u; u; Au/ D 0. The relation (7.13) is called the enstrophy equation. Exercise 7.21. Prove that for the two-dimensional Navier–Stokes flow with periodic boundary conditions we have not only b.u; u; u/ D 0 but also b.u; u; Au/ D 0, cf. [197, 220]. The latter property makes the analysis of the problem much easier. From (7.13) we obtain d kuk2 C jAuj2 Cjf j2 ; dt and, as kuk C1 jAuj for u 2 D.A/, we have also d kuk2 C k1 kuk2 Cjf j2 : dt

(7.14)

Exercise 7.22. Prove that kuk C1 jAuj for u 2 D.A/ and compute the constants C, C1 , and k1 above. Now, to estimate the norm ku.t/k we cannot use the Gronwall inequality since our initial condition u0 is only in H and not in V. Instead, we shall use the uniform Gronwall inequality (cf. Lemma 3.21). To apply the uniform Gronwall lemma, we proceed as follows. First, from inequality (7.11), after integration in the interval .t; t C r/, where t t0 .u0 /, we obtain in particular, Z tCr ku.t/k2 ds C./rjf j2 C 2 D a3 (7.15) t

for all t t0 .B1 /. Then, from inequality (7.14) and the uniform Gronwall lemma we get directly

a 3 C a2 for all t t0 C r; (7.16) ku.t/k2 r where Z

tCr

a2 D t

Cjf j2 dt D rCjf j2 ;

a3 D

1 .rCjf j2 C 2 /;

[cf. (7.15)]. Estimate (7.16) means that our bounded set B1 is moved by S.t/, for t t0 C r, into a ball in V,

a 3 C a2 for all t t0 C r; x 2 B1 ; kS.t/xk2 r

7.3 Convergence to the Stationary Solution: The Simplest Case

157

which is a precompact set in H; due to the Rellich lemma, V is compactly embedded in H. Thus, the whole trajectory of the set B1 in H, starting from t0 C r, [

S.t/B1

is precompact in H:

tt0 Cr

This is the condition of the uniform compactness of the semigroup S.t/ in H. From the above considerations and Theorem 7.3 it follows the existence of the global attractor A H for the two-dimensional Navier–Stokes equations with periodic boundary conditions. t u Exercise 7.23. Prove the existence of a global attractor in the case of homogeneous Dirichlet boundary conditions. To deal with the trilinear form b.u; u; Au/ apply the Agmon inequality kuk1 c2 juj1=2 jAuj1=2

u 2 D.A/:

for

Exercise 7.24. Prove that from the uniform compactness of the semigroup S.t/ in H it follows the existence of a compact absorbing set in H. Exercise 7.25. Prove that if f 2 L1 .RC I H/, then the bounded absorbing set in V exists for the corresponding non-autonomous problem.

7.3 Convergence to the Stationary Solution: The Simplest Case Let us recall the stationary problem: u C .u r/u C rp D f

in Q;

div u D 0

in Q;

with one of the boundary conditions: 1. Q is a square Œ0; L2 in R2 and we assume periodic boundary conditions, or 2. Q is a bounded domain in R2 , with smooth boundary, and we assume the homogeneous boundary condition u D 0 on @Q. Let H and V be the corresponding function spaces, cf. Sect. 4.1. Then the weak formulation of the above problem is as follows. Problem 7.2. For f 2 H (or f 2 V 0 ) find u 2 V such that ..u; v// C b.u; u; v/ D hf ; vi

for all

v2V

158


or, equivalently, Au C Bu D f

in H

or

V 0:

Assuming that the viscosity is large enough with respect to the volume force f we shall prove convergence of solutions u.t/ of the nonstationary problem to the unique stationary solution u1 as time goes to infinity. Theorem 7.5. There exists a constant c1 .Q/ such that if 2 > c1 .Q/kf kV 0 then the stationary solution u1 is unique. Moreover, for each u0 2 H the solution u of the problem: 2 .0; 1I V/; u 2 C.Œ0; 1/I H/ \ Lloc

u.0/ D u0 ;

d .u.t/; v/ C ..u.t/; v// C b.u.t/; u.t/; v/ D hf ; vi for all dt

v 2 V;

converges to u1 in H as time goes to infinity, ju.t/ u1 j ! 0 as

t ! 1:

Proof. Let w.t/ D u.t/ u1 . Since Au1 C B.u1 / D f , we have dw.t/ C Aw.t/ C B.u.t// B.u1 / D 0 dt

in V 0 :

Thus, in particular,

dw.t/ C Aw.t/ C B.u.t// B.u1 /; w.t/ D 0; dt

that is, 1d jw.t/j2 C kw.t/k2 C b.u.t/; u.t/; w.t// b.u1 ; u1 ; w.t// D 0: 2 dt Further, using a property of the trilinear form b, we obtain 1d jw.t/j2 C kw.t/k2 D b.w.t/; u1 ; w.t//: 2 dt

(7.17)

We shall estimate the right-hand side. Using the Ladyzhenskaya inequality we obtain b.u; v; w/ kukL4 .Q/2 kvkkwkL4 .Q/2 c1 juj1=2 kuk1=2 kvkjwj1=2 kwk1=2 :

7.4 Convergence to the Stationary Solution for Large Forces

159

Thus, b.w.t/; u1 ; w.t// c1 jw.t/jku1 kkw.t/k c2 kw.t/k2 C jw.t/j2 ku1 k2 : 2 We have then d c2 jw.t/j2 C kw.t/k2 jw.t/j2 ku1 k2 ; dt and further, as jw.t/j Ckw.t/k, we have

c2 d jw.t/j2 C C ku1 k2 jw.t/j2 0: dt

(7.18)

Now, we know that ku1 k 1 kf kV 0 , whence C

c2 c2 ku1 k2 C 3 kf k2V 0 > 0

for large [satisfying 2 > c1 .Q/kf kV 0 for some constant c1 .Q/]. Thus, from inequality (7.18) it follows (for these ) the uniqueness of the stationary solution u1 and also the exponential decay in H of u.t/ to u1 as t ! 1, and this ends the proof. t u Exercise 7.26. Prove the above theorem without using the Ladyzhenskaya inequality, by estimating in a different way the right-hand side of Eq. (7.17). Compare the constants c1 .Q/ in both cases for the problem with periodic boundary conditions.

7.4 Convergence to the Stationary Solution for Large Forces We consider two-dimensional flows on Q D Œ0; L2 as in the preceding sections , where ˛ > 0 and prove that for any force of the form f D 0; ˛ sin 2 x 1 L can be arbitrarily large, all nonstationary solutions of the Navier–Stokes problem converge to the unique solution uN D 1 1 f of the stationary Navier–Stokes problem, where 1 D

4 2 L2

is the first eigenvalue of the Stokes operator.

Remark 7.5. Observe that for our force f the Grashof number GD

jf j Lj˛j Dp ; 2 1 2 2 1

160


which is a dimensionless quantity which measures the strength of the forcing relative to viscosity, is large for large ˛. It is easy to see that f has the following properties, div f D 0;

Af D 1 f ;

B.f ; f / D 0:

(7.19)

Let us write the energy equation 1d 2 juj C kuk2 D .f ; u/; 2 dt and the enstrophy equation 1d kuk2 C jAuj2 D .f ; Au/ D .Af ; u/ D 1 .f ; u/: 2 dt We have used the property, cf. Exercise 7.21, b.u; u; Au/ D 0: Multiplying the energy equation by 1 and subtracting it from the enstrophy equation we have 1d .kuk2 1 juj2 / C .jAuj2 1 kuk2 / D 0: 2 dt Observe that kuk2 1 juj2 0 in view of the Poincaré inequality. More precisely, recalling that uD

1 X

uO k !k

kD1

we have, kuk2 1 juj2 D

1 X

.k 1 /jOuk j2 ;

kD5

and also jAuj2 1 kuk2 D

1 X

k .k 1 /jOuk j2

kD5

5

1 X .k 1 /jOuk j2 D 5 .kuk2 1 juj2 /: kD5

7.4 Convergence to the Stationary Solution for Large Forces

161

Thus, we have 1d .kuk2 1 juj2 / C 5 .kuk2 1 juj2 / 0; 2 dt and lim .ku.t/k2 1 ju.t/j2 / D 0:

t!1

(7.20)

Now, let u D P4 u C Q4 u. Then jQ4 uj2 D

1 X

jOuk j2

kD5

D

1 X 1 .k 1 /jOuk j2 5 1 kD5

1 .kuk2 1 juj2 /: 5 1

(7.21)

Therefore, from (7.20) and (7.21) we conclude that lim jQ4 u.t/j2 D 0:

t!1

We have shown that all higher harmonics go to zero as time goes to infinity. Let uN D 1 1 f be the stationary solution. We define v.t/ D u.t/ uN , the difference between some nonstationary solution and the stationary solution uN . As P4 uN D uN then Q4 v.t/ D Q4 u.t/ ! 0 as t ! 1. We shall show that also P4 v.t/ ! 0 as t ! 1, so that v.t/ ! 0 and hence u.t/ ! uN as t ! 1. We have dv C Av C B.v; v/ C B.v; uN / C B.Nu; v/ D 0: dt We multiply this equation by P4 v in H and use kP4 vk2 D 1 jP4 vj2 to get 1d jP4 vj2 C 1 jP4 vj2 C b.v; v; P4 v/ C b.v; uN ; P4 v/ C b.Nu; v; P4 v/ D 0: 2 dt Let us write v D P4 v C Q4 v and use jQ4 v.t/j ! 0 as t ! 1. We can write the above equation in the form 1 d jP4 vj2 C 1 jP4 vj2 C b.P4 v; P4 v; P4 v/ C b.P4 v; uN ; P4 v/ C b.Nu; P4 v; P4 v/ D ˇ.t/; 2 dt

where ˇ.t/ ! 0 as t ! 1 since all terms in ˇ.t/ are of the form b.x; y; z/ where at least one variable is equal to Q4 v.t/. Moreover b.P4 v; P4 v; P4 v/ D 0 and b.P4 v; uN ; P4 v/ C b.Nu; P4 v; P4 v/ D b.P4 v; P4 v; uN / D 0:

162


Exercise 7.27. Write P4 v as a sum of the first four eigenvectors to check directly that b.P4 v; P4 v; uN / D 0. In this way we have obtained 1d jP4 vj2 C 1 jP4 vj2 ˇ.t/; 2 dt

ˇ.t/ ! 0;

t ! 1;

whence lim jP4 u.t/j2 D 0;

t!1

and lim ju.t/ uN j D 0:

t!1

This relation proves also the uniqueness of the stationary solution. Exercise 7.28. Write ˇ.t/ explicitly and prove that ˇ.t/ ! 0 as t ! 1. Exercise 7.29. Assume that the external force f satisfies: B.f ; f / D 0 and Af D m f for m large enough. Prove that the stationary solution uN D 1m f is unique and globally stable, that is, all nonstationary solutions u.t/ converge to uN in H as t ! 1. Exercise 7.30. Assume that the external force f D fm C fn satisfies: B.fm ; fm / D 0, B.fn ; fn / D 0, B.fn ; fm / D 0 and Afm D m fm , Afn D n fn for m ; n large enough. Find the unique stationary solution uN and prove that it is globally stable, that is, all nonstationary solutions u.t/ converge to uN in H as t ! 1. Generalize the result to any finite combination f D fm1 C : : : C fmk . Solution to Exercise 7.29. Multiplying the equation du C Au C B.u; u/ D f dt by m and using m f D Af we obtain m

du C A.m u f / C m B.u; u/ D 0: dt

Let v.t/ D m u.t/ f (then u D t ! 1. Observe that m

1 .v m

C f /). We shall show that v.t/ ! 0 as

1 dv du D ; dt dt

hence, multiplying Eq. (7.22) by we have the following equation for v.t/, dv C Av C B.u; u/ D 0; dt

(7.22)

where

uD

1 .v C f /: m

7.5 Average Transfer of Energy

163

Writing the nonlinear term in terms of v, and using B.f ; f / D 0 we obtain dv 1 C Av C .B.v; v/ C B.v; f / C B.f ; v// D 0: dt m Now, multiplying both sides by v in H we get 1 1d 2 jvj C kvk2 C b.v; f ; v/ D 0: 2 dt m Since Af D m f then kf k2 D m jf j2 , and ˇ ˇ p ˇ ˇ 1 ˇ c1 kvk2 kf k D c1 kvk2 m jf j D pc1 kvk2 jf j; ˇ b.v; f ; v/ ˇ ˇ m m m m whence, 1d 2 c1 jvj C p jf j kvk2 0: 2 dt m

(7.23)

For any fixed f satisfying (7.19) there exists a natural number m0 such that for all m m0 , m D pc1 jf j > 0 and then all solutions v.t/ converge to zero in m

H at least exponentially fast as t ! 1. In fact, using jvj2 from (7.23), jv.t/j2 jv.0/j2 exp.21 m t/

for

1 kvk2 1

we obtain

t 0:

This means that u.t/ ! uN D 1m f in H as t ! 1 and proves also that uN is the unique solution of the stationary problem. Remark 7.6. Observe that though the norm of our f in H may be large, its norm in H 1 is equal to p1 jf j and for m large is small. m

7.5 Average Transfer of Energy We shall consider the Navier–Stokes system as in Sect. 7.1. Let e.u/ D 12 juj2 be the kinetic energy of the flow. From the Navier–Stokes equation we obtain d e.u.t// D ku.t/k2 C .f ; u.t// dt and we can see that, thanks to the identity b.u; u; u/ D 0, the global balance of the kinetic energy of the flow does not depend on the nonlinear term .u r/u. It is

164


the same as in the case of the linear (Stokes) flow. In this section we shall study more precisely the transport of energy in the fluid. We know (cf. Sect. 7.4) that the velocity u.x; t/ can be represented as a Fourier series u.x; t/ D

1 X

uO k .t/!k .x/;

kD1

where !k , k D 1; 2; 3; : : : are eigenfunctions of the Stokes operator constituting an orthonormal family in the phase space H. We have thus, 1

e.u/ D

1X jOuk j2 : 2 kD1

Let u D y C z, where yD

m X

uO k !k ;

kD1

zD

1 X

uO k !k :

kDmC1

We have, e.u/ D e.y/ C e.z/; where 1X jOuk j2 ; 2 kD1 m

e.y/ D

e.z/ D

1 1 X jOuk j2 2 kDmC1

are kinetic energies of the large scale component y of the flow and the small scale component z. Now, in the balance of the kinetic energies of components y and z the nonlinear term will not reduce to zero. We shall show how the kinetic energy of particular components of the flow is distributed in time among the Fourier modes. Let us assume that the external force is given by f D

m2 X

fOk !k ;

1 m1 m2 < 1:

kDm1

Let Pm W u ! y, Qm W u ! z be orthogonal projections, with Qm D I Pm . Applying these projections to the Navier–Stokes system du C Au C B.u; u/ D f ; dt

7.5 Average Transfer of Energy

165

and assuming that m m2 ; we obtain dy C Ay C Pm B.y C z; y C z/ D Pm f dt and dz C Az C Qm B.y C z; y C z/ D 0; dt as Qm z D 0 (m m2 ), Pm A D APm , Qm A D AQm . The corresponding energy equations are 1d 2 jyj C kyk2 C b.y C z; y C z; y/ D .f ; y/; 2 dt and 1d 2 jzj C kzk2 C b.y C z; y C z; z/ D 0: 2 dt Define E.u/ D kuk2 —the enstrophy. Using the property b.u; v; w/ D b.u; w; v/ we can write d e.y/ C E.y/ D ˚z .y/ C .f ; y/; dt

(7.24)

d e.z/ C E.z/ D ˚y .z/; dt

(7.25)

and

where ˚z .y/ D b.z; z; y/ C b.y; y; z/;

˚y .z/ D b.y; y; z/ C b.z; z; y/:

Observe that ˚z .y/ C ˚y .z/ D 0. In (7.24) ˚z .y/ is the net amount of kinetic energy per unit time transferred from small length scales z to large length scales y (similarly, ˚y .z/ is the net amount of kinetic energy per unit time transferred from large length scales y to small length scales z), while the term E.y/ represents the rate of energy dissipation by viscous effects per unit time for the large length scale and .f ; y/ is the energy flow injected into the large length scales by the forcing term.

166


We shall show that on average, over time and space, ˚y .z/ 0, that is, if the external forces act in the range of large length scales y then, on average, the net transfer of kinetic energy occurs only into the small scales.

This transfer of energy is called direct energy transfer. To prove the above statement, we integrate Eq. (7.25) in t to get Z e.z.T// C

Z

T

E.z.s//ds D

0

0

T

˚y .z.s//ds C e.z.0//:

As the function t ! e.z.t// is bounded for t 0, dividing both sides by T and passing with T to infinity we obtain, in particular, 0 lim inf T!1

1 T

Z

T

E.z.s//ds D lim inf T!1

0

1 T

Z

T

0

˚y .z.s//ds:

Assuming that the limit Z

1 lim T!1 T

T 0

˚y .z.s//ds

exists, we have 1 1 lim j˝j T!1 T

Z

T

0

˚y .z.s//ds 0:

Exercise 7.31. Let us assume that the external force is given by f D

m2 X

fOk !k ;

kDm1

where now 1 m m1 , m1 > 1. Thus, the force acts in the range of small length scales. Prove that the transfer of energy is then from higher modes (small length scales) to lower modes (large length scales), which is called an inverse energy transfer. The inverse energy transfer in the large scales may be responsible for the observed persistent low-wave number coherent structures in turbulent flows.

7.6 Comments and Bibliographical Notes In this chapter we introduced and applied some basic tools and techniques of dealing with the nonstationary Navier–Stokes equations. As in the case of the stationary Navier–Stokes equations, the problem of existence of solutions was resolved by introducing the appropriate notion of a weak solution. We proved the uniqueness of a


167

weak solution and considered, in the simplest context, a problem of regularity of the weak solution. We saw the utility of the Galerkin method of approximate solutions, the importance of compactness theorems, here, the Aubin–Lions compactness theorem, in particular. The usefulness of basic functional inequalities: the Poincaré inequality, the Ladyzhenskaya inequality, and the Agmon inequality, as well as the role of the Gronwall and the uniform Gronwall inequalities were demonstrated. Further considerations and details concerning the existence of solutions and global attractors for autonomous two-dimensional Navier–Stokes flows in a bounded domain can be found, e.g., in [88, 197, 220]. For the study of threedimensional case see, e.g., [76]. For the existence and regularity of solutions for problems on unbounded domains which do not satisfy the Poincaré inequality see [237, 238] and for the problems on cylindrical domains see [192, 196, 233]. In Sects. 7.4 and 7.5 we followed closely the argument from [99]. In Chap. 8 we shall show the connections of the global attractor with statistical solutions of the Navier–Stokes equations. In Chap. 9 we shall apply the presented techniques to a problem of a shear flow taken from the theory of lubrication and shall prove that the global attractor has a finite fractal dimension. Non-autonomous system of the Navier–Stokes equations and flows in unbounded domains will be considered in Chaps. 11 and 13, respectively.

8

Invariant Measures and Statistical Solutions

In this chapter we prove the existence of invariant measures associated with two-dimensional autonomous Navier–Stokes equations. Then we introduce the notion of a stationary statistical solution and prove that every invariant measure is also such a solution.

8.1 Existence of Invariant Measures We shall consider flows with homogeneous Dirichlet boundary conditions, in a smooth bounded domain ˝. Let f 2 H. Then there exists a semigroup fS.t/gt0 in H such that the solution of the problem ut C Au C B.u; u/ D f ;

u.0/ D u0

is given by u.t/ D S.t/u0 for t 0. The map .t; u/ ! S.t/u is continuous from Œ0; 1/ H to H. Our first aim is to prove the following theorem. Theorem 8.1. Let LIMT!1 be a fixed Banach generalized limit. Then for every u0 2 H there exists a probability measure on H such that LIMT!1

1 T

Z

T 0

Z .S.t/u0 /dt D

.u/d .u/

(8.1)

H

for every continuous function W H ! R ( 2 C.H/). Moreover, the measure is regular, invariant with respect to the semigroup fS.t/gt0 , that is, for every -measurable set E H it holds .S.t/1 E/ D .E/, and its support is contained in the global attractor A . © Springer International Publishing Switzerland 2016 G. Łukaszewicz, P. Kalita, Navier–Stokes Equations, Advances in Mechanics and Mathematics 34, DOI 10.1007/978-3-319-27760-8_8

169

170

8 Invariant Measures and Statistical Solutions

We shall explain the notion of a Banach generalized limit LIMT!1 . Definition 8.1. LIMT!1 —a Banach generalized limit is any linear functional on the set of all bounded functions W Œ0; 1/ ! R ( 2 B.Œ0; 1//), such that • LIMT!1 .T/ 0 for • LIMT!1 .T/ D limt!1

0; .T/ if the limit on the right-hand side exists.

Remark 8.1. Such functionals exist, there may be many of them, hence in Eq. (8.1) the measure depends not only on the initial condition u0 but also on the choice of LIMT!1 [note that in (8.1) the right-hand side is defined by the left-hand side]. To prove the existence of LIMT!1 we shall use the Hahn–Banach theorem (cf. Theorem 8.2). Let G0 D f

2 B.Œ0; 1// W limT!1 .T/

existsg:

The map 0 . / D limT!1 .T/ is a linear functional defined on the subspace G0 of B.Œ0; 1//. We can extend it to a linear functional defined on the whole space B.Œ0; 1//. In fact, let p. / D lim supt!1 .t/. Then 0 . / p. /

on

G0 ;

and p.

C / p. / C p./;

p.˛ / D ˛p. /;

˛0

for ; 2 B.Œ0; 1//. In view of the Hahn–Banach theorem there exists an extension of 0 to the whole space, satisfying . / p. / D lim supt!1 .t/ for all 2 B.Œ0; 1//. It then follows (cf. also Exercise 8.1) that . / 0 for all nonnegative 2 B.Œ0; 1//. We denote . / D LIMT!1 .T/. 2 B.Œ0; 1//,

Exercise 8.1. Prove that for all

lim inf .T/ LIMT!1 .T/ lim sup .T/; 1 T!1

T!1

whence jLIMT!1 .T/j lim sup .T/ sup j .T/j: T!1

T0

We are going now to prove Theorem 8.1. At first, we assume that there exists a measure as in (8.1) for some fixed u0 2 H and LIMT!1 , and derive its basic properties which follow from this formula. 1. is a probability measure, i.e., .H/ D 1. Take 1 on H to get 1 D LIMT!1

1 T

Z 0

T

Z 1dt D

1d .u/ D .H/: H

8.1 Existence of Invariant Measures

171

2. is an invariant measure for the semigroup fS.t/gt0 . We shall prove that Z

Z .S.t/u/d .u/ D H

.u/d .u/

(8.2)

H

for all 2 C.H/. To this end observe that by the uniform compactness of the semigroup (cf. Sect. 7.2) we have Z .S. /u/d .u/ D LIMT!1 H

1 T

Z

T 0

.S.t C /u0 /dt

Z 1 TC .S.t/u0 /dt T Z n1 Z T 1 TC D LIMT!1 .S.t/u0 /dt C .S.t/u0 /dt T 0 T T Z o 1 .S.t/u0 /dt T 0 Z Z 1 T .S.t/u0 /dt D .u/d .u/: D LIMT!1 T 0 H D LIMT!1

This identity can be extended to all functions which are -integrable on H. Setting D E —the characteristic function of E H, we obtain Z

Z E .S.t/u/d .u/ D H

E .u/d .u/; H

which can be written as .S.t/1 E/ D .E/. Thus, is invariant. Exercise 8.2. Let identity (8.2) hold for all 2 C.H/, be a regular measure [in the sense of Theorem 8.3 (c)] and C.H/ be a dense subset of L1 .H; d /. Prove that then (8.2) holds for all functions which are -integrable on H. Prove that the latter property is equivalent to the property of invariance of given by .S.t/1 E/ D .E/. Remark 8.2. To prove invariance of , the uniform compactness property is not needed. In the argument above we need only the boundedness of the function t ! .S.t/u0 /, which follows directly from the existence of the global attractor. In fact, if .S.t/u0 / is not bounded on the positive semiaxis t 0, then there exists a sequence sn ! 1 such that j.S.sn /u0 /j n

for each

n 1:

(8.3)

But then, as distH .S.t/u0 ; A / ! 0 as t ! 1, we can find a subsequence s0n and a sequence vn , vn 2 A , such that distH .S.s0n /u0 ; vn / 1=n for each n 1. Then, as A is compact, there is a subsequence of the vn (which we relabel) such that

172


vn ! v 2 A . But then also S.s0n /u0 ! v, and therefore .S.s0n /u0 / ! .v/ as m ! 1, which contradicts (8.3). 3. The support of is contained in the global attractor A . We shall make use of the following lemma. Lemma 8.1. The support of the measure contains only the nonwandering points, that is, points u 2 H such that for every open neighborhood A of u in H there exists a sequence of times tn ! 1 for which S.tn /A \ A 6D ; for every n. Remark 8.3. The set M1 of nonwandering points is closed and invariant. If the phase space is compact, then M1 is a nonempty and compact set. Then, every trajectory converges to M1 [186]. Proof (of Lemma 8.1). Let F be the support of the measure , F D supp —the smallest closed set such that .F/ D 1. Step 1. First we shall prove that for u 2 F and a set A containing u which is relatively open in F, .A/ > 0. Assuming that .A/ D 0 we obtain .FnA/ D 1, F n A is closed, compact in F, and different from F, hence F is not the support of which gives a contradiction. Fig. 8.1 Illustration of the set C .A / n S.T/ C .A / used in the proof of Lemma 8.1

Step 2. To prove that u 2 F is nonwandering, let us assume to the contrary, that for some T S and all t T, S.t/A \ A D ;. Consider the positive trajectory C C .A/ D t0 S.t/A of the set A. By assumption, A \ S.T/ .A/ D ;. Moreover, S.T/ C .A/ C .A/, A C .A/ n S.T/ C .A/, (see Fig. 8.1) and using step 1 we get 0 < .A/ < . C .A/ n S.T/ C .A// D . C .A// .S.T/ C .A//: On the other hand, .S.T/ C .A// D .S.T/1 S.T/ C .A// . C .A//; as always S.T/1 S.T/B B. Thus, we have come to a contradiction. The proof of Lemma 8.1 is complete. t u


173

Corollary 8.1. The support of the measure is contained in the global attractor A for the semigroup fS.t/gt0 . Proof. Every point outside of the global attractor is wandering, that is, it is not a nonwandering point. u t We have also the following property of the measure . Corollary 8.2. .S.t/E/ D .E/ for all -measurable E A and t 2 R. Proof. The semigroup fS.t/gt0 reduced to A is a complete dynamical system (we have a result about the backward uniqueness on the attractor for the twodimensional Navier–Stokes equations, see [197], Theorem 12.8), hence, in view of the invariance of , .E/ D .S.t/1 S.t/E/ D .S.t/E/; t u

which ends the proof. Proof of the existence of the measure in (8.1)

Step 1. We shall prove that for every u0 2 H and for every continuous function W H ! R there exists a limit Z 1 T .S.t/u0 /dt: L./ D LIMT!1 T 0 Assume we have chosen the functional LIMT!1 . It suffices to show that the function Z 1 T f .T/ D .S.t/u0 /dt T 0 is bounded on the interval Œ0; 1/. We know that there exists a compact absorbing set X for the semigroup fS.t/gt0 (cf. Sect. 7.2). Hence, the function t ! .S.t/u0 / is continuous and bounded on the compact set C .u0 /. Therefore, the function T ! .T/ is bounded (and also continuous) on Œ0; 1/. Step 2. L./ depends only on values of on any compact absorbing set X. It means that if 1 is another function in C.H/ which is equal to on X then L./ D L.1 /. To prove this property observe that there exists T0 such that .S.t/u0 / 1 .S.t/u0 / D 0 for t T0 , as S.t/u0 2 X for large times. Thus, Z 1 T ..S.t/u0 / 1 .S.t/u0 //dt L. 1 / D LIMT!1 T 0 Z 1 T0 ..S.t/u0 / 1 .S.t/u0 //dt D 0: D LIMT!1 T 0

174


Step 3. Consider a compact absorbing set X as above. From the Riesz–Kakutani representation theorem (Theorem 8.3) we know that for every positive linear continuous functional G on the space of continuous functions on X (G 2 C.X/0 ), there exists a Borel measure on X such that Z G. / D

.u/d .u/

for all

2 C.X/0 :

X

We can extend this measure to the whole phase space setting .E/ Q D .E \ X/ for E H. We shall denote this extended measure simply by in what follows. Then, in particular, .H n X/ D 0. Further, from the Tietze extension theorem (Theorem 8.4) we know that every function 2 C.X/ can be extended to a continuous function on the whole space without enlarging the norm. Thus, 1 LIMT!1 T

Z

T 0

.S.t/u0 /dt D L./ D G.jX /

.L./ depends only on jX /

D G. / .jX D / Z D .u/d .u/ .Riesz–Kakutani theorem/ Z

X

D

.u/d .u/

.support of is contained in X/:

H

Since is a time averaged measure, it is invariant. We have shown that every such measure is supported on the global attractor. Thus, .A / D 1 and LIMT!1

1 T

Z

T 0

.S.t/u0 /dt D

1 .A /

The proof of Theorem 8.1 is complete.

Z A

.u/d .u/ for all

2 C.H/: t u

Remark 8.4. The existence of a compact absorbing set is not needed to prove the existence of . We shall provide a simple proof that works in uniformly convex Banach spaces. Let K be the closed convex hull of the global attractor. As A is compact, it is known that K is also a compact subset of H (see [3], Theorem 5.35). As K is compact and H is uniformly convex, for each u 2 H there exists a unique ku 2 K such that ju ku j D infk2K ju kj and the projection operator P W u 2 H ! ku 2 K is continuous. Given u0 2 H consider t ! P.S.t/u0 /, the projection onto K of the trajectory t ! S.t/u0 . Since K is compact, the function Œ0; 1/ 3 t ! .P.S.t/u0 // 2 R is continuous and bounded for 2 C.H/.


175

Moreover j.S.t/u0 / .P.S.t/u0 //j ! 0 as

s ! 1:

(8.4)

.P.S.t/u0 //dt:

(8.5)

We then conclude that LIMT!1

1 T

Z

T 0

.S.t/u0 /dt D LIMT!1

1 T

Z 0

T

The right-hand side defines a linear positive functional on C.K/. The rest of the proof is similar to that in step 3 above, where now K plays the role of the compact absorbing set X. Exercise 8.3. Prove that the projection operator P W u 2 H ! ku 2 K is continuous. Exercise 8.4. Prove (8.4) and (8.5). Below we present the basic theorems which have been used above. Theorem 8.2 (Hahn–Banach). Let X be a real linear space, p W X ! R a function such that p.x C y/ p.x/ C p.y/;

p.˛x/ D ˛p.x/

for ˛ 0, x; y 2 X. Let f be a real-valued linear functional on a subspace Y of X such that f .x/ p.x/ for x 2 Y. Then there exists a real linear functional F on X such that F.x/ D f .x/;

x 2 Y;

F.x/ p.x/;

x 2 X:

Theorem 8.3 (Riesz–Kakutani). Let X be a locally compact Hausdorff space. Let be a positive linear functional on Cc .X/, the space of real-valued continuous functions of compact support on X. Then there exist a -algebra M on X containing all Borel sets in X and a unique positive measure on M such that: R (a) f D X f .u/d .u/ for every f 2 Cc .X/, (b) .K/ < 1 for every compact set K X, (c) is regular in this sense that for every E M , .E/ D inff ./ W E ;

is openg;

and, for every E 2 M with .E/ < 1, .E/ D supf .K/ W K E; (d)

K is compactg:

is complete, i.e., if E 2 M , A E and .A/ D 0, then A 2 M .

176


Theorem 8.4 (Tietze). Let X be a normal topological space and A be a closed subset of X. If f is a continuous and bounded real-valued function defined on A, then there exists a continuous real-valued function F defined on X such that: (i) F.x/ D f .x/ for x 2 A, (ii) supx2X jF.x/j D supx2X jf .x/j.

8.2 Stationary Statistical Solutions Let u.t/ D S.t/u, t 2 RC , be a family of maps in .X; 0 /. If t .E/ D 0 .S.t/1 E/ for all 0 -measurable sets E, then Z Z ˚.u/d t .u/ D ˚.S.t/u/d 0 .u/ for all ˚ 2 L1 .X; t /: (8.6) X

X

If du.t/ D F.u.t//; dt

(8.7)

then from (8.6) by formal differentiation with respect to t and from (8.7) we obtain Z Z d d ˚.u/d t .u/ D ˚.S.t/u/d 0 .u/ dt X dt X Z Z d 0 ˚ .S.t/u/ .S.t/u/d 0 .u/ D ˚ 0 .S.t/u/F.S.t/u/d 0 .u/ D dt X X Z ˚ 0 .u/F.u/d t .u/: D X

Hence, d dt

Z

Z ˚.u/d t .u/ D X

˚ 0 .u/F.u/d t .u/:

(8.8)

X

The above equation we shall call the Liouville equation. If measure 0 is invariant with respect to the semigroup of transformations fS.t/gt0 , then 0 .T 1 E/ D 0 .E/ D t .E/; that is, all measures t , t 0, coincide with 0 . In this case the Liouville equation (8.8) reduces to the stationary Liouville equation. Z

˚ 0 .u/F.u/d .u/ D 0; X

where we denoted by the measure t D 0 .

(8.9)

8.2 Stationary Statistical Solutions

177

Let us consider the two-dimensional Navier–Stokes equations du.t/ D F.u.t//; dt u.0/ D u0 :

F.u/ D f Au B.u/;

We proved (in Sect. 8.1) existence of probability measures invariant with respect to the semigroup fS.t/gt0 associated with the above Navier–Stokes system. Thus, there exists a Borel measure on the phase space H for which .S.t/1 E/ D .E/ for Borel sets E in H. Moreover, .H/ D 1. For such measure formula (8.9) holds true. We shall give the precise meaning to this formula in the case of the twodimensional Navier–Stokes dynamical system. Let us define the class of the test functions ˚. Definition 8.2. To the class T of test functions belong real-valued functionals ˚ D ˚.u/ defined on H, which are bounded on bounded subsets of H and such that (i) For any u 2 V, the Fréchet derivative ˚ 0 .u/ taken in H along V exists. More precisely, for each u 2 V, there exists an element in H denoted ˚ 0 .u/ such that j˚.u C v/ ˚.u/ .˚ 0 .u/; v/j ! 0 as jvj

jvj ! 0;

v 2 V;

(ii) ˚ 0 .u/ 2 V for all u 2 V, and u ! ˚ 0 .u/ is continuous and bounded as a function from V to V. For example, P let ˚.u/ D ..u; g1 /; : : : ; .u; gn //, where ˚ 0 .u/ D njD1 @j ..u; g1 /; : : : ; .u; gn //gj 2 V. Let denote a probability measure in H such that Z

2 C01 .Rn /, gj 2 V. Then

kuk2 d .u/ < 1;

(8.10)

H

where kuk D 1 for u 2 H n V. We thus have .H n V/ D 0, that is, the support of is included in V, and the map u ! .Au C B.u/ f ; ˚ 0 .u// is continuous in V for every test function ˚ 2 T . Moreover, j.Au C B.u/ f ; ˚ 0 .u//j D j..u; ˚ 0 .u/// C b.u; u; ˚ 0 .u// .f ; ˚ 0 .u//j kukk˚ 0 .u/k C ckuk2 k˚ 0 .u/k C jf jj˚ 0 .u/j 1=2

.kuk C ckuk2 C 1 jf j/ sup k˚ 0 .u/k: u2V

(8.11)

178


As .HnV/ D 0, from (8.10) and (8.11) it follows [cf. condition (ii) of the definition of the class T ] that the integral Z .Au C B.u/ f ; ˚ 0 .u//d .u/ H

has sense and is finite. Now we can define the stationary statistical solution of the two-dimensional Navier–Stokes system. Definition 8.3. Stationary statistical solution of the Navier–Stokes system is a probability measure on H such that R (i) RH kuk2 d .u/ < 1, (ii) H .F.u/; ˚ 0 .u//d .u/ D 0 for all ˚ 2 T , where F.u/ D Au C B.u/ f , Rf 2 H, (iii) E1 juj2
Z

Z .f ; u/d .u/ jf j

2

jujd .u/ jf j

E

E

juj d .u/

12

1

.E/ 2

E

jf j

Z

1

12

kuk2 d .u/

12

< 1:

E

Then from (iii), Z

2

Z

kuk d .u/ E

.f ; u/d .u/ E

jf j 1

12

Z

kuk d .u/ E

and we have, Z

kuk2 d .u/

jf j2 ; 2 1

juj2 d .u/

jf j2 : 2 21

E

and Z E

2

The last inequality we can write in the form Z jf j2 2 juj 2 2 d .u/ 0: 1 E

12

< 1;

8.2 Stationary Statistical Solutions

For E1 D

jf j2 2 21

179

and E2 D 1 we have juj2

jf j2 0 2 21

for all u 2 E, whence .E/ D 0. Thus jf j u 2 H W juj D 1; 1 and

jf j supp u 2 H W juj 1

\ V:

Now, we shall prove existence of a stationary statistical solution for the twodimensional Navier–Stokes system. Theorem 8.5. Let be a probability measure on H invariant with respect to the semigroup fS.t/gt0 associated with the two-dimensional Navier–Stokes system. Then is a stationary statistical solution of the system. Step 1. Since is invariant, it is concentrated on the global attractor. Thus

Proof.

Z

Z

2

kuk2 d .u/ < 1;

kuk d .u/ D A

H

and we have the first property of the stationary statistical solution. Step 2. We shall show that Z .F.u/; ˚ 0 .u//d .u/ D 0;

(8.12)

H

where F.u/ D Au C B.u/ f , F.u/ D ut , ˚ 2 T . From the invariance of , for every ˚ 2 T , Z

.F.u/; ˚ 0 .u//d .u/ D

H

Z

.F.S.t/u/; ˚ 0 .S.t/u//d .u/;

H

and the right-hand side is independent of t 0. Therefore, from the Fubini theorem, Z

.F.S.t/u/; ˚ 0 .S.t/u//d .u/ D

H

1 T Z

D H

Z

T 0

1 T

Z

Z 0

.F.S.t/u/; ˚ 0 .S.t/u//d .u/dt H T

.F.S.t/u/; ˚ 0 .S.t/u//dtd .u/:

180


Thus, Z

Z

0

.F.u/; ˚ .u//d .u/ D H

H

1 f˚.S.t/u/ ˚.u/gd .u/; T

(8.13)

as d ˚.u/ D .F.u/; ˚ 0 .u//: dt The right-hand side of (8.13) tends to 0 as T ! 1, as the function t ! ˚.S.t/u/ is bounded. This proves (8.12). Step 3. Now we shall prove that Z E1 juj2
fkuk2 .f ; u/gd .u/ D 0

(8.14)

for all 0 E1 < E2 1. In particular the left-hand side is not positive which gives the third property of the stationary statistical solution. From the energy equation 1 jS.t/uj2 C 2

Z

t

kS.s/uk2 ds D

0

1 2 juj C 2

Z 0

t

.f ; S.s/u/ds;

we obtain, by regrouping the terms, Z

t 0

fkS.s/uk2 .f ; S.s/u/gds D

1 fjuj2 jS.t/uj2 g 2

(8.15)

for t 0. From the invariance of , the Fubini theorem, and (8.15) we get Z E1 juj2
fkuk2 .f ; u/gd .u/ D

D D D

1 T Z

Z

T 0

Z E1 juj2
E1 juj2
1 2

Z

1 T

E1 juj2
Z

0

T

Z E1 juj2
fkS.s/uk2 .f ; S.s/u/gd .u/

fkS.s/uk2 .f ; S.s/u/gd .u/ fkS.s/uk2 .f ; S.s/u/gd .u/

1 fjuj2 jS.t/uj2 gd .u/ T

for every T 0. Passing with T to 1 we obtain zero on the right-hand side. This proves (8.14). t u


181

Remark 8.5. In fact, the converse is also true. If is a stationary statistical solution, then is a probability measure invariant with respect to the semigroup fS.t/gt0 (cf. Chap. 4 in [99]).

8.3 Comments and Bibliographical Notes Our presentation follows that in [99], where this subject is treated in greater detail. The alternative proof of Theorem 8.1 and its generalization to noncompact semigroups presented in Remarks 8.2 and 8.4 come from [165]. Theorem 8.5 follows also from the results of Chap. 12, cf. [160]. For statistical solutions for the Navier–Stokes equations, see [96, 97, 99, 101, 116, 195, 203, 228]. Some recent results for three-dimensional problems can be found in [29, 104].

9

Global Attractors and a Lubrication Problem

We start this chapter from necessary background on the theory of fractal dimension. Next, we formulate and study a problem which models the two-dimensional boundary driven shear flow in lubrication theory. After the derivation of the energy dissipation rate estimate and a version of Lieb–Thirring inequality we provide an estimate from above on the global attractor fractal dimension.

9.1 Fractal Dimension We start this section from the definition and some properties of a fractal dimension of a relatively compact set K X, where X is a Banach space. By N"X .K/ we denote the minimal number of closed balls in X of radius " needed to cover the set K. From the relative compactness of K it follows that this number is finite. Obviously, we expect the quantity N"X .K/ to increase as " ! 0. The fractal dimension, given by the following definition, measures how fast is this increase. Definition 9.1. Let K X be a relatively compact set. The fractal dimension of K, denoted by dfX .K/ is a number defined by dfX .K/ D lim sup "!0

log N"X .K/ : log 1"

The value dfX .K/ can be equal to 1 even if K is a compact set. If the set K is not relatively compact, we set dfK .K/ D 1. In the next result we recall several properties of dfX .


183

184

9 Global Attractors and a Lubrication Problem

Theorem 9.1 (Cf. [197, Proposition 13.2]). (i) if Kk X for k D 1; : : : ; N are relatively compact sets, then for every k D 1; : : : ; N dfX .Kk /

dfX

N [

! Kk

max dfX .Kk /:

kD1

kD1;:::;N

(ii) If f W X ! X is Hölder continuous with the exponent 2 .0; 1, i.e., kf .x/ f .y/kX Lkx yk X

for all

x; y 2 X;

then dfX .f .K//

dfX .K/

for every relatively compact

K X:

(iii) We have dfX .K/ D dfX .K/ for every relatively compact K X, where K is the closure of K in X. Note that property (i) does not hold for countable unions. Indeed, let X D R and K D Œ0; 1 \ Q. From (iii) it follows that dfX .K/ D dfX .Œ0; 1/, and it is easy to verify that dfX .Œ0; 1/ D 1, while for every point x 2 K we have dfX .fxg/ D 0. If we know that an infinite dimensional semiflow fS.t/gt0 is defined by the solution maps of an initial and boundary value problem and it has a global attractor A , then we can restrict the dynamical system to the attractor, an invariant set, i.e., treat the family fS.t/gt0 as the mappings S.t/ W A ! A , thus interpreting the global attractor as the image of the flow after a long time of evolution (keeping in mind that all trajectories are arbitrarily close to the attractor after a long time, but each trajectory does not necessarily have one corresponding trajectory on the attractor, to which it becomes attracted in the norm, such trajectories exist, but only on finite time intervals, the length of which goes to infinity as time goes to infinity, see [197] Proposition 10.14 and Corollary 10.15). An important question that arises is the following: can the dynamics on the attractor be described by means of the time evolution of a finite number of variables without loss of information? From the fact that the global attractor exists, we only know that it is compact, and hence it must have an empty interior in an infinite dimensional phase space H, but such compact sets can be still “large.” Indeed, if a Banach space V embeds compactly in the phase space H, then any set, that is bounded in V, is relatively compact in H, so, potentially, a ball in V is an admissible candidate for the attractor. In certain cases, however, it is possible to obtain much more knowledge on the attractor than only its compactness: namely it is possible to show that it has finite fractal dimension. If we could prove that the attractor is a finite dimensional manifold, then it would be natural to use its parameterization, and reduce the dynamics to the finite dimensional one. Unfortunately, the set of finite

9.2 Abstract Theorem on Finite Dimensionality and an Algorithm

185

fractal dimension can be very complicated and it does not have to be a manifold. Still, due to the Hölder–Mañé Theorem, if the fractal dimension of the attractor is finite we can reduce the dynamics on the attractor to the finite dimensional one. Theorem 9.2 (Hölder–Mañé, cf. [117]). Let H be a Banach space and let A H be a compact set such that dfH .A / m2 . Then, almost every bounded linear function W H ! Rm is a bijection as a map W A ! .A / and its inverse 1 j.A / is Hölder continuous with a certain exponent ˛ 2 .0; 1/ depending only on m and dfH .A /. The expression “almost every” in the last theorem is understood in the sense of prevalence, a generalization on the notion “almost everywhere in the sense of Lebesgue measure” to the infinite dimensional Banach spaces. Here we consider the prevalence in the space of linear and bounded operators L .HI Rm /. Definition 9.2. The Borel set E X, where X is a Banach space, is said to be prevalent if there exists a compactly supported probability measure such that .E C x/ D 1 for all x 2 X. A non-Borel set that contains a prevalent set is also prevalent. Using a linear map that satisfies the assumptions of Theorem 9.2 it is possible to construct the finite dimensional semiflow S .t/ W .A / ! .A / for t 0 conjugate with the original one according to the formula S .t/x D .S.t/ 1 x/. Unfortunately, this new semiflow is not necessarily Lipschitz continuous, even if S.t/ are Lipschitz, since 1 is only Hölder. Therefore it can have more than one solution. Still, this result plays an important role in the theory of finite dimensional reduction of infinite dimensional dynamical systems, and, clearly, the finite dimensionality of the global attractor is a strong evidence that the semiflow is in fact finite dimensional. More information about this theory can be found in the monograph [198], and, in context of two-dimensional Navier–Stokes equations, in the review article [199].

9.2 Abstract Theorem on Finite Dimensionality and an Algorithm In this section we present two classes of methods that can be used to prove the finite dimensionality of the global attractor. The first of this two approaches was introduced by Constantin and Foia¸s in the article [74] (also see [57, 58, 220]) and it relies on the linearization of the underlying PDEs and the fact that if the linearized system contracts m-dimensional volumes uniformly on the global attractor, then its fractal dimension must not exceed m. Below we present an algorithm, based on this idea, that can be used to obtain the estimate of the fractal dimension of the global attractor for the Navier–Stokes equations. The exposition is based on [57, 220].

186


Let fS.t/gt0 be a semiflow on a Hilbert space X, that has a global attractor A , being a compact, invariant, and attracting set in X. Definition 9.3. We say that the semiflow fS.t/gt0 is uniformly quasidifferentiable on A if for every t 0 and u 2 A there exists a linear and continuous operator .t; u/ 2 L .XI X/ such that for all t 0 sup

sup

u;v2A 0
kS.t/v S.t/u .t; u/.v u/kX ! 0 as kv ukX

! 0:

(9.1)

Note (see Sect. V.2.2 in [220]) that the mapping .t; u/ does not have to be defined uniquely. We assume that the semiflow is given by a solution map of the following abstract evolution problem. du.t/ D F.u.t// in X dt u.0/ D u0 in X;

for t > 0;

where F W W ! X, W being a certain Banach space continuously embedded in X. Then we have u.t/ D S.t/u0 . We construct the following linearized problem dU.t/ D F 0 .S.t/u0 /U.t/ in X dt U.0/ D in X;

for

t > 0;

(9.2) (9.3)

where F 0 is the Fréchet derivative of F, and we make the following assumption. Assumption A. The semiflow fS.t/gt0 is uniformly quasidifferentiable on A , with sup sup k.t; u/kL .XIX/ < 1:

(9.4)

t2Œ0;1 u2A

Moreover, for u0 2 A the mapping X Œ0; 1/ 3 .; t/ ! .t; u0 / D U.t/ 2 X is a solution map of the linearized system (9.2)–(9.3). In practice, the verification of this, technical, assumption depends on the structure of the studied problem and relies on the construction of the linearized problem (9.2)– (9.3) and proving that its solution map is a quasidifferential, i.e., it satisfies (9.1), and that (9.4) holds (see, for instance, examples in Sects. VI.2.1 and VI.3.1 in [197]). In addition to Assumption A, for the fractal dimension of a global attractor to be finite (and equal to d), we need the linearized semiflow to contract uniformly the d-dimensional volumes. To understand this notion properly we must first recall several useful ideas.


187

If 1 ; : : : ; m 2 X, then the tensor product 1 ˝ 2 ˝ : : : ˝ m is an m-linear form over X defined as .1 ˝ 2 ˝ : : : ˝ m /.

1; : : : ;

m/

D .1 ;

1 /X .2 ;

2 /X

: : : .m ;

m /X ;

m elements from and the space spanned Nm by all tensor products 1 ˝ 2 ˝ : : : ˝ m ofN m X is denoted by X. We will be interested in the subspace of X denoted by Vm X D X ^ X ^ : : : ^ X (X appears m times on the right-hand side). This subspace, containing the m-linear antisymmetric forms, is spanned by the forms 1 ^ 2 ^ : : : ^ m being the exterior products of the elements from X, defined by 1 ^ 2 ^ : : : ^ m D

X

sgn. /.1/ ˝ .2/ ˝ : : : ˝ .m/ ;

where the sum is taken over all permutations V of the set f1; : : : ; mg, and sgn. / is the sign of the permutation . We define on m X the inner product given, for exterior products of elements from X, by the formula .1 ^ 2 ^ : : : ^ m ;

1

^

2

^ ::: ^

m/

Vk

X

D detf.i ;

j /X gi;jD1;:::;m ;

V and we extend it to the whole k X by linearity. The associated norm is given, for exterior products of elements from X, by k1 ^ 2 ^ : : : ^ m k2Vk X D detf.i ; j /X gi;jD1;:::;m : The importance of the norm of the exterior product k1 ^ 2 ^ : : : ^ m kVm X lies in the fact that this quantity is the volume of the m-dimensional parallelepiped with the edges given by 1 ; : : : ; m . We will start the evolution from the volume spanned by the vectors which are the initial conditions of the linearized system (9.2)–(9.3) and we will derive the condition which guarantees that this volume is uniformly contracted on the attractor by the linearized semiflow. To this end, for a linear V and bounded operator L 2 L .XI X/ let us introduce the operator Lm W X m ! m X by the formula Lm .1 ; : : : ; m / D L1 ^ 2 ^ : : : ^ m C 1 ^ L2 ^ : : : ^ m C : : : C 1 ^ 2 ^ : : : ^ Lm :

188


We have the following lemma. Lemma 9.1 (Cf. [220, Lemma V.1.2]). For any 1 ; : : : ; m 2 X we have .Lm .1 ; : : : ; m /; 1 ^ : : : m /Vm X D k1 ^ 2 ^ : : : ^ m k2Vm X Tr.L ı Q/; where Q is the orthogonal projection on the space spanned by fi gm iD1 and Tr is the trace of the operator of the finite rank, given by m X

i

i L Tr.L ı Q/ D ; ; k i kX k i kX X iD1 m where f i gm 1D1 is the orthogonal basis of the space spanned by fi giD1 , obtained, for example, by the Gram–Schmidt procedure.

We fix u0 2 A and we take the linearly independent vectors 1 ; : : : ; m 2 H. Next, we solve (9.2)–(9.3) for U1 .t/; : : : ; Um .t/ and calculate how the volume spanned by U1 .t/; : : : ; Um .t/ is related to the volume spanned by 1 ; : : : ; m (see Fig. 9.1). We have

Fig. 9.1 Transformation by a linearized semiflow .t; u0 / of a two-dimensional volume spanned by 1 ; 2

1d kU1 .t/ ^ : : : ^ Um .t/k2Vm X 2 dt d D .U1 .t/ ^ : : : ^ Um .t//; U1 .t/ ^ : : : ^ Um .t/ V m dt X D .U10 .t/ ^ : : : ^ Um .t/; U1 .t/ ^ : : : ^ Um .t//Vm X C : : : C .U1 .t/ ^ : : : ^ Um1 .t/ ^ Um0 .t/; U1 .t/ ^ : : : ^ Um .t//Vm X D .F 0 .S.t/u0 /U1 .t/ ^ : : : ^ Um .t/; U1 .t/ ^ : : : ^ Um .t//Vm X C : : :


189

C .U1 .t/ ^ : : : ^ Um1 .t/ ^ F 0 .S.t/u0 /Um .t/; U1 .t/ ^ : : : ^ Um .t//Vm X D ..F 0 .S.t/u0 //m .U1 .t/; : : : ; Um .t//; U1 .t/ ^ : : : ^ Um .t//Vm X D kU1 .t/ ^ : : : ^ Um .t/k2Vm X Tr.F 0 .S.t/u0 / ı Q.t//; where Q.t/ D Q.t; u0 I 0 ; : : : ; m / is the orthogonal projection onto the space spanned by U1 .t/; : : : ; Um .t/. It follows that kU1 .t/ ^ : : : ^ Um .t/kVm X D k1 ^ : : : ^ m kVm X e.

Rt 0

Tr.F 0 .S.t/u0 /ıQ.t// dt/

:

We introduce the quantity !m .S.t/u0 / D

sup 1 ;:::;m 2X;ki kX 1

kU1 .t/ ^ : : : ^ Um .t/kVm X ;

being the largest volume obtained from the m-dimensional parallelepiped with sides no greater than one, by the linearized semiflow. We also introduce auxiliary quantities ! m .t/ D sup !m .S.t/u0 /; u0 2A

1 qm .t/ D sup sup u0 2A 1 ;:::;m 2X;ki kX 1 t whence we have

1 t

Z 0

t

Tr.F 0 .S.t/u0 / ı Q.t// dt;

log ! m .t/ qm .t/. Finally, we introduce the numbers qm D lim sup qm .t/: m!1

It is easy to deduce the following result Proposition 9.1. If, for certain m 2 NC , qm < 0; then, for some t0 > 0 and all t t0 we have qm .t/ ı < 0, and the volume element kU1 .t/ ^ : : : ^ Um .t/kVm X decays exponentially as t ! 1 uniformly for u0 2 A and 1 ; : : : ; m 2 X, i.e., kU1 .t/ ^ : : : ^ Um .t/kVm X ceıt

for t t0 :

We conclude the presentation of the method with a result that links the values qm with the fractal dimension of the global attractor. Theorem 9.3 (Cf. [57, Corollary 2.2]). Let Assumption A hold and let f W Œ0; 1/ ! R be a concave function such that for every positive natural number m we have qm f .m/ and let d > 1 be such that f .d / D 0. Then

190


dfX .A / d : The above result gives rise to an algorithm. We need to estimate from above the quantities Tr.F 0 .S.t/u0 / ı Q.t//; and by taking the supremum over 1 ; : : : ; m and over the points u0 in the attractor, after the integration over time, we obtain the bound on qm by the concave function of m. Then, it is enough to find a zero of this function to get the bound on the global attractor fractal dimension. We conclude with two remarks: firstly, the algorithm works only if the phase space is a Hilbert space, and secondly, the semiflow must be smooth, i.e., it must be possible to linearize the underlying initial and boundary value problem and the solution map of the linearized problem must be a quasidifferential of the original semiflow in accordance with Definition 9.3. Further in this chapter we will apply the described algorithm to estimate the attractor dimension for a shear flow in the lubrication theory. Before we move to this problem, however, we give an account of other techniques useful for the fractal dimension estimation. The second class of methods useful for proving the attractor finite dimensionality relies on obtaining the estimates on the difference of two trajectories. We present several results from this class of methods. Theorem 9.4 (Cf. [168, 236]). Let E be a bounded subset of a Banach space H. Let moreover V be another Banach space compactly embedded in H and let L W H ! H be a mapping such that E L.E/, and we have kLu LvkV cku vkH

u; v 2 E

for all

with c > 0:

(9.5)

Then, the fractal dimension of E is finite and dfH .E/

log N H1 .BV .0; 1// 4c

log 2

;

where BV .0; 1/ is the closed unit ball in V. Proof. We choose R > 0 and u 2 E such that E BH .u; R/ (we use the notation BH .x; R/ D fu 2 H W ku xkH Rg and BV .x; R/ D fu 2 V W ku xkV Rg). We have R ; BH vi ; E L.E/ BV .Lu; cR/ 4 iD1 N [


191

where vi 2 H and N is the number of balls in H having the radius R4 needed to cover the ball in V centered at zero having the radius cR. Note that N D N H1 .BV .0; 1//. 4c

We can assume that all balls from the last sum have a nonempty intersection with E, and we denote the arbitrary points from these intersections by zi , whence R E ; BH zi ; 2 iD1 N [

with zi 2 E. It follows that ED

N [ iD1

R \E : BH zi ; 2

Now R R \ E BV Lzi ; c : BH zi ; 2 2 The ball in V having the radius c R2 can be covered by N balls in H having the radius R , whence 8 N [ R R \E ; BH zi ; BH vij ; 2 8 jD1 and, if we want the centers to be the points of E, we have N [ R R \E ; BH zi ; BH zij ; 2 4 jD1 with zij 2 E. Repeating this procedure inductively, we can construct a cover of E by R N k balls of radius 2Rk . For " > 0 we can always find k 2 N such that 2kC1 < " 2Rk , whence N" .E/ N R .E/ N kC1 , and 2kC1

log N" .E/ .k C 1/ log N : k log 2 log R log 1" The assertion follows.

t u

To get the finite dimensionality of the global attractor by the above theorem, it would seem easiest to apply it for L D S.t/ for certain t > 0 and E D A . It is then enough to derive the estimate (9.5), which is known as the smoothing estimate as it implies that in fact we have A V. While this estimate may be difficult to

192


obtain for L D S.t/, in some cases it is easier to get it for L being the translation operator defined on the trajectories starting from the attractor (and staying in it, by invariance), and V; H being the appropriately chosen spaces of time-dependent functions. This method, known as the method of l-trajectories, was introduced in [168] and will be used for the two-dimensional Navier–Stokes equations with nonmonotone friction in Sect. 10. The next result, very useful for proving finite dimensionality of global attractors of infinite dimensional dynamical systems, proved originally by Ladyzhenskaya (see, for example, [147], where, however, the notion of the Hausdorff dimension is used), uses the so-called squeezing property. This property involves the estimates which are not, as in Theorem 9.4, in the space compactly embedded in the state space, but which involve the finite dimensional projectors. The variant of Ladyzhenskaya result, that we present here, comes from [70]. Theorem 9.5 (Cf. [70, Theorem 8.1]). Let H be the Hilbert space and let E H be a compact set. Let moreover L W H ! H be a continuous mapping such that E L.E/ and there exists an orthogonal projector PN W H ! HN onto the subspace HN H of dimension N, such that kP.Lu Lv/kH lku vkH

for all

u; v 2 E;

and k.I P/.Lu Lv/kH ıku vkH

for all

u; v 2 E;

where ı < 1. Then the fractal dimension dfH .E/ is finite and dfH .E/

9l N log 1ı

log

2 1ı

1 :

The above theorem has also the non-autonomous version (see Theorem 13.5) which will be used later in Chap. 13 to prove the finite dimensionality of the pullback attractor for two-dimensional boundary driven Navier–Stokes flow. Comments We refer the reader to [198] for more information about dimensions. Fractal dimension is also known as upper box counting dimension (see [198] Definition 3.1). Sometimes another notion of dimension, namely, the above mentioned Hausdorff dimension, is used to study the global attractors for infinite dimensional dynamical systems. To define it, one needs to use the s-dimensional Hausdorff measure valid for s 0 and a subset K of a Banach space X, given by ( HsX .K/ D lim inf ı!0

1 X iD1

jUi js W K

1 [ iD1

) Ui with jUi j ı ;

9.3 An Application to a Shear Flow in Lubrication Theory

193

where jAj D supx;y2A kx ykX . We define the Hausdorff dimension as dHX .K/ D inffs 0 W HsK .K/ D 0g: For the properties of dHX see Chap. 2 in [198]. Here we only note that, in contrast to the fractal dimension, the Hausdorff dimension is stable under the countable unions (see Proposition 2.8 in [198]), and for a set K X, we have dHX .K/ dfX .K/; whence the ability to estimate from above the fractal dimension gives us also the corresponding estimate on the Hausdorff dimension, while the opposite implication does not hold (in particular it is possible to construct sets with zero Hausdorff dimension and infinite fractal dimension, see [198], Exercise 13.3). We also note that both the notions of fractal and Hausdorff dimensions remain valid in metric spaces.

9.3 An Application to a Shear Flow in Lubrication Theory In this section we consider a two-dimensional Navier–Stokes shear flow for which there exists a unique global in time solution as well as the global attractor for the associated semigroup. Our aim is to estimate from above the fractal dimension of the attractor in terms of given data and geometry of the domain of the flow. The problem we consider is motivated by a typical problem from lubrication theory where the domain of the flow is usually very thin and the Reynolds number of the flow is large.

9.3.1 Formulation of the Problem We consider two-dimensional Navier–Stokes equations, ut u C .u r/u C rp D 0;

(9.6)

div u D 0;

(9.7)

in the channel ˝1 D fx D .x1 ; x2 / W 1 < x1 < 1; 0 < x2 < h.x1 /g; where h is a function, positive, smooth, and L-periodic in x1 .

194


Let ˝ D fx D .x1 ; x2 / W 0 < x1 < L; 0 < x2 < h.x1 /g; and @˝ D C [ L [ D , where C and D are the bottom and the top, and L is the lateral part of the boundary of ˝. We are interested in solutions of (9.6)–(9.7) in ˝ which are L-periodic with respect to x1 . Moreover, we assume u D 0 on

D ;

u D U0 e1 D .U0 ; 0/

(9.8) C ;

on

(9.9)

where U0 is a positive constant, and the initial condition u.x; 0/ D u0 .x/

for

x 2 ˝:

(9.10)

The problem is motivated by a flow in an infinite (rectified) journal bearing ˝ .1; C1/, where D .1; C1/ represents the outer cylinder, and C .1; C1/ represents the inner, rotating cylinder. In the lubrication problems the gap h between cylinders is never constant. We can assume that the rectification does not change the equations as the gap between cylinders is very small with respect to their radii. Remark 9.1. When h D const, problem (9.6)–(9.10) was intensively studied in several contexts, cf. Sect. 9.4. In our considerations we use the background flow method and transform the boundary condition (9.9) to the homogeneous ones by defining a smooth background flow, a simple version of the Hopf construction, which we shall be described in detail. Let u.x1 ; x2 ; t/ D U.x2 /e1 C v.x1 ; x2 ; t/;

(9.11)

with U.0/ D U0 ;

U.h.x1 // D 0;

x1 2 .0; L/:

(9.12)

Then v is L-periodic in x1 and satisfies vt v C .v r/v C Uv;x1 C.v/2 U 0 e1 C rp D U 00 e1 ; div v D 0; v D 0 on

D [ C ;

(9.13) (9.14) (9.15)


195

and the initial condition v.x; 0/ D v0 .x/ D u0 .x/ U.x2 /e1 :

(9.16)

By .v/2 we denoted the second component of v. Now, we define a weak form of the transformed problem above. To this end we need some notations. Let VQ D fv 2 C1 .˝/2 W v is L-periodic in x1 ; div v D 0; v D 0 on D [ C g; V D closure of VQ in H 1 .˝/2 ; H D closure of VQ in L2 .˝/2 : We define the scalar products Z .u; v/ D

˝

u.x/ v.x/dx;

and ..u; v// D .ru; rv/ in H and V, respectively, and the corresponding norms 1

jvj D .v; v/ 2

and

1

kvk D ..v; v// 2 :

Let b.u; v; w/ D ..u r/v; w/; and a.u; v/ D .ru; rv/: Then the natural weak formulation of the transformed problem (9.13)–(9.16) is as follows. Problem 9.1. Find v 2 C.Œ0; TI H/ \ L2 .0; TI V/ for each T > 0, such that d .v.t/; / C a.v.t/; / C b.v.t/; v.t/; / D F.v.t/; / dt

(9.17)

196


for all 2 V, and v.x; 0/ D v0 .x/; where F.v; / D a.; / b.; v; / b.v; ; /;

(9.18)

and D Ue1 is a suitable background flow. We have the following existence theorem. Theorem 9.6. There exists a unique weak solution of Problem 9.1 such that for all , T, 0 < < T, v 2 L2 .; T I H 2 .˝//, and for each t > 0 the map v0 7! v.t/ is continuous as a map in H. Moreover, there exists a global attractor for the associated semigroup fS.t/gt0 in the phase space H. Proof. The standard existence part of the proof is based on an energy inequality, the Galerkin approximations, and the compactness method [220]. We shall not reproduce the standard argument, however, we stop for a moment to mention the properties of the Stokes operator and the regularity problem. As usual, we define the Stokes operator as the self-adjoint operator from V to its dual V 0 and also as an unbounded strictly positive operator in H with compact inverse A1 , by hAu; vi D a.u; v/

for all

u; v 2 V:

Then, there exists a complete orthonormal family wj , j 2 N in H, with wj 2 D.A/ D fu 2 V W Au 2 Hg H 2 .˝/, such that Awj D j wj , 0 < 1 2 : : :, j ! 1 as j ! 1. The inclusion D.A/ H 2 .˝/ can be proved in the usual way by using the regularity theory. First, we extend u; p, and f by periodicity to some smooth set ˝3 such that ˝ ˝3 ˝1 , and then use the classical Cattabriga–Solonnikov estimate [219] to the solution .'u; 'p/ of the related generalized Stokes problem in the domain ˝3 , with ' being a suitable cut-off function that equals one on ˝. This argument gives us also the equivalence of the norms kukH 2 .˝/ and on D.A/. PjAuj m Now, defining the Galerkin approximations by vm .x; t/ D v.t/w Q j .x/ we jD1 2 can see that they belong to H .˝/ for t > . Further estimates performed on the Galerkin approximations lead, in a standard way, to v 2 L2 .; TI H 2 .˝//. t u Exercise 9.1. Provide the details of the proof of Theorem 9.6.

9.3.2 Energy Dissipation Rate Estimate Our aim now is to estimate the time averaged energy dissipation rate per unit mass of the flow u—the weak solution of problem (9.6)–(9.10). We define


D

1 < kuk2 > D lim sup j˝j T!C1 j˝j T

197

Z

T

ku.t/k2 dt:

(9.19)

0

We estimate first the averaged energy dissipation rate of the flow v, and then use the relation, cf. (9.11), Z Z 2 2 0 ku.t/k D kv.t/k C 2 U .v/1 ;x2 dx C jU 0 j2 dx: (9.20) ˝

˝

To estimate the right-hand side of (9.20) we use Eq. (9.17). Taking D v in (9.17), we obtain 1d 2 jvj C a.v; v/ C b.v; v; v/ D F.v; v/: 2 dt Since v D 0 on D [ C , is L-periodic in x1 , and div v D 0, we have b.v; v; v/ D 0, and 1d 2 jvj C kvk2 D F.v; v/: 2 dt

(9.21)

Integrating the above equation in t on the interval .0; T/, dividing by T, and taking lim sup of both sides we estimate the averaged energy dissipation rate of v. Before, however, we have to estimate carefully the term F.v; v/ on the right-hand side of (9.21). By (9.18), F.v; v/ D a.; v/ b.v; ; v/:

(9.22)

We have ja.; v/j kkkvk kk2 C

kvk2 : 4

(9.23)

To estimate the last term in (9.22) we use the following lemma. Lemma 9.2. For any > 0 there exists a smooth extension D .x2 / D U.x2 /e1 D .U.x2 /; 0/ of the boundary condition for u, such that jb.v; ; v/j kvk2

for all

v 2 V:

Proof. Let W Œ0; 1/ ! Œ0; 1 be a smooth function such that .0/ D 1;

supp Œ0; 1=2;

max j0 .s/j

p

8;

198


and let, for 0 < " 1, U.x2 / D U0 .x2 =.h0 "//;

h0 D min h.x1 /: 0x1 L

Then the function U matches the boundary conditions (9.12), is L-periodic in x1 , and div Ue1 D 0. Moreover, for Q" D .0; L/ .0; h0 "=2/, v krvkL2 .Q" / kx2 U.x2 /kL1 .Q" / : jb.v; Ue1 ; v/j D jb.v; v; Ue1 /j x 2 2 L .Q" / Now, kx2 U.x2 /kL1 .Q" /

h0 " U0 ; 2

and ˇ ˇ 2 Z L Z h.x1 / ˇ Z L Z h.x1 / ˇ ˇ v.x1 ; x2 / ˇ2 ˇ @v.x1 ; x2 / ˇ2 v ˇ ˇ ˇ ˇ dx2 dx1 ˇ x ˇ dx2 dx1 4 ˇ @x ˇ x 2 2 L .Q" / 2 2 0 0 0 0 by the Hardy inequality (cf. Theorem 3.12 and Exercise 3.12). Thus, we obtain jb.v; Ue1 ; v/j kvk2

with D 2h0 U0 ":

t u

In particular, jb.v; ; v/j

kvk2 4

for " D minf=.8h0 U0 /; 1g:

(9.24)

In view of estimates (9.23) and (9.24), jF.v; v/j kk2 C

kvk2 ; 2

and, by (9.21), 1d 2 jvj C kvk2 kk2 : 2 dt 2

(9.25)

To estimate the right-hand side in terms of the data, we use Lemma 9.3. Let U be as in Lemma 9.2. Then Z 1 jU.x2 /j2 dx1 dx2 Lh0 U02 "; 2 ˝

(9.26)


199

and Z ˝

jU 0 .x2 /j2 dx1 dx2

4LU02 1 : h0 "

(9.27)

Proof. We have jU.x2 /j U0 and supp U Q" , which gives (9.26). Moreover, Z

as j0 j

p

ˇ Z ˇ ˇ 0 x2 ˇ2 ˇ dx ˇ ˇ h0 " ˇ ˝ ˇ Z h0 " ˇ 2 ˇ U02 4LU02 1 x2 ˇˇ2 0 ˇ ; D 2 2L dx 2 ˇ h0 " ˇ h0 " h0 " 0

U2 jU .x2 /j dx D 2 02 h0 " ˝ 0

2

8, which gives the second inequality of the lemma.

t u

Applying Lemma 9.3 to inequality (9.25) we obtain 4LU02 1 1d 2 jvj C kvk2 : 2 dt 2 h0 "

(9.28)

Let hM D max0x1 L h.x1 / and hM =h0 ' 1. Then we can define the Reynolds number of the flow u by Re D .h0 U0 /=. Observe that for a turbulent flow u (of some large Reynolds number) " in (9.24) is equal to =.8h0 U0 /. Therefore, from (9.28) we obtain, as indicated above, Lemma 9.4. If Re 1 then the time averaged energy dissipation rate per unit mass .v/ for the flow v can be estimated as follows: .v/

64L 3 64 3 U U : j˝j 0 h0 0

In the end, we have the following theorem. Theorem 9.7. For the Navier–Stokes turbulent flow u with Re 1 the time averaged energy dissipation rate per unit mass defined in (9.19) can be estimated as follows: C

U03 ; h0

(9.29)

where C is a numeric constant. Proof. By (9.20) we have 2

2

< kuk > 2 < kvk > C2

Z

jU 0 j2 dx; ˝

and then we use (9.27) to estimate the second term on the right-hand side.

t u

200


Estimate (9.29) has the same form as the usual in turbulence theory estimate of the averaged energy dissipation rate for the Navier–Stokes boundary driven flow in a rectangular domain [87, 88, 99]. In the sequel we shall use the estimates of to find an upper bound of the dimension of the global attractor.

9.3.3 A Version of the Lieb–Thirring Inequality Let Q 1 D fv 2 C1 .˝/2 W v is L -periodic in x1 ; v D 0 on D [ C g H and Q 1 in H 1 .˝/2 : H 1 D closure of H Lemma 9.5. Let 'j 2 H 1 , j D 1; : : : ; m be an orthonormal family in L2 .˝/ and let hM D max0x1 L h.x1 /. Then the following inequality holds: Z

0 12 2 ! X m m Z X hM 2A @ 'j dx 1 C jr'j j2 dx; L ˝ ˝ jD1 jD1

(9.30)

where is an absolute constant. Proof. Let ˝1 D .0; L/ .0; h0 /, and let j 2 H 1 .˝1 /, j D 1; : : : ; m, be a family of functions that are orthonormal in L2 .˝1 /, with j .0; x2 / D j .L; x2 / and j D 0 for x2 D 0, x2 D h0 . We know [89] that for this family there exist absolute constants C00 and C10 , such that Z ˝1

0 m X @

12

0 m Z X 2A 0 @ dx C j 0

jD1

jD1

1 12 0 1 12 ˇ ˇ ˇ ˇ2 m Z 0 X ˇ @ j ˇ2 ˇ ˇ C1 m A @ ˇ ˇ ˇ @ j ˇ dxA : ˇ @x ˇ dx C L2 ˇ @x ˇ 1 2 ˝1 jD1 ˝1

We have also the Poincaré inequality Z ˝1

2 j dx

h2M

ˇ ˇ ˇ @ j ˇ2 ˇ ˇ dx: ˇ ˇ ˝1 @x2

Z

From the above two inequalities and the observation that mD

m Z X jD1

(9.30) follows.

˝1

2 j dx;

t u


201

9.3.4 Dimension Estimate of the Global Attractor We rewrite Eq. (9.17) for the transformed flow v in the short form dv D L.v/; dt

v.0/ D v0 ;

where, for all 2 V, hL.v.t//; i D a.v.t/; / b.v.t/; v.t/; / C F.v.t/; /; and F is defined in (9.18). Now, to estimate from above the dimension of the global attractor we follow the procedure described in Sect. 9.2. First, for an integer m > 1 and vectors j 2 H, j D 1; : : : ; m, we consider the corresponding linearized around the orbit v.t/ problems d Uj D L0 .v/Uj dt

and

Uj .0/ D j

for

j D 1; : : : ; m;

(9.31)

where L0 .v/ is the Fréchet derivative of L at v D v.t/, with .L0 .v/Uj ; / D a.Uj ; / b.v; Uj ; / b.Uj ; v; / b.; Uj ; / b.Uj ; ; / D a.Uj ; / b.u; Uj ; / b.Uj ; u; /:

(9.32)

Let, for a particular time , j D j . /, j 2 N be an orthonormal basis of H with 1 . /; : : : ; m . / spanning Qm . /H D Qm .; v0 ; 1 ; : : : ; m /H D spanfU1 . /; : : : ; Um . /g; Qm . / being the orthogonal projector of H onto the space spanned by Uj . / for j D 1; : : : ; m, solutions of (9.31). We then have j . / 2 V, j D 1; : : : ; m, for a.e. 2 RC . The trace of L0 .v. // ı Qm . / is m 0 X Tr L0 .v. // ı Qm . / D L .v. //j . /; j . / :

(9.33)

jD1

Let A be the global attractor for the transformed flow v./ D S. /v0 , and let qm D lim sup qm .T/; T!1

202


where qm .T/ D sup

v0 2A

sup j 2H; jj j1;jD1;:::;m

1 T

Z

T 0

0 Tr L .v. // ı Qm . / d :

Now, our aim is to obtain estimate (9.37) and then inequality (9.38). Lemma 9.6. The following estimate holds: m X Tr L0 .v/ ı Qm kj k2 C jjL2 .˝/ kuk;

(9.34)

jD1

where .x/ D

m X

jj .x/j2 :

jD1

Proof. From (9.32) and (9.33) we obtain m m m X X X 0 L .v/j ; j kj k2 b.j ; u; j / jD1

jD1

m X jD1

jD1

kj k2 C

Z

0 jruj @

˝

m X

1 jj .x/j2 A dx;

jD1

t u

whence (9.34) follows.

Now, in our estimates of the trace we use Lemma 9.5. By this lemma, taking into account that hM L for a thin domain ˝, and from Z .x/dx D m ˝

we have X m2 jj2L2 .˝/ 1 kj k2 ; j˝j jD1 m

(9.35)

where 1 D 2 . Moreover, m 1 X 1 2 2 jjL2 .˝/ kuk jj 2 C kj k2 C kuk kuk2 21 L .˝/ 2 2 jD1 2

(9.36)


203

by the second inequality in (9.35). Thus, by (9.34), (9.36) m X 1 Tr L0 .v/ ı Qm kj k2 C kuk2 ; 2 jD1 2

and by (9.35) again, we obtain in the end Tr L0 .v. // ı Qm . /

1 m2 C kuk2 : 21 j˝j 2

(9.37)

From (9.37), recalling defined in (9.19), we have qm

1 j˝j m2 C : 21 j˝j 2 2

(9.38)

Define aD

; 21 j˝j

cD

1 j˝j : 2 2

We can write qm am2 C c: By Theorem 9.3 we have dfH .A / p; where p > 0 is such that ap2 C c D 0. Obviously, we have r pD

c : a

Further, taking into account definitions of a and c, as well as estimate (9.29) of , we obtain p 1

j˝j .Re/3=2 : h20

(9.39)

From the above estimates we conclude the following theorem about strongly turbulent flows in thin (or elongated) domains.

204


Theorem 9.8. Consider the Navier–Stokes flow u as described in this section. Assume that the domain ˝ is thin and that the flow is strongly turbulent, namely hM

1 L

and

Re 1:

(9.40)

Then the fractal dimension of the global attractor A for this flow can be estimated as follows: dfH .A /

j˝j .Re/3=2 ; h20

(9.41)

where is some absolute constant. For a rectangular domain ˝ D .0; L/ .0; h0 / we obtain, in particular, dfH .A /

L .Re/3=2 : h0

Proof. By (9.39), (9.40), and by the definition of m0 , (9.41) follows.

t u

9.4 Comments and Bibliographical Notes The standard procedure of estimating the global attractor dimension based on the dynamical system theory [88, 99, 220], we used in Sect. 9.3, involves two important ingredients: estimate of the time averaged energy dissipation rate and a Lieb– Thirring-like inequality. The precision and physical soundness of an estimate of the number of degrees of freedom of a given flow (expressed by an estimate of its global attractor) depends directly on the quality of the estimate of and a good choice of the Lieb–Thirring-like inequality which depends, in particular, on the geometry of the domain and on the boundary conditions of the flow. There is a quickly growing literature devoted to better and better estimates of averaged parameters and attractor dimension of a variety of flows. We mention only a few positions which are related to the problem considered in Sect. 9.3 and some other to give a larger context. Estimate (9.41) is a direct generalization of that in [89], where the domain of the flow is an elongated rectangle ˝ D .0; L/ .0; h/, L h. In this case the attractor dimension can be estimated from above by c Lh Re3=2 , is the Reynolds number. Here, the where c is a universal constant, and Re D Uh dependence on the geometry of the domain is explicit. (To obtain such result the authors prove a suitable anisotropic form of the Lieb–Thirring inequality. For a variety of forms of this inequality cf. [220]. Moreover, a discussion on agreement of their estimates with Kolmogorov and Kraichnan scaling in turbulence theory is provided in [89].)


205

In [241] optimal bounds of the attractor dimension are given for a flow in a rectangle .0; 2L/ .0; 2L=˛/, with periodic boundary conditions and given external forcing. The estimates are of the form c0 =˛ dfH .A / c1 =˛, see also [180]. A free boundary conditions are considered in [242], see also [222], and an upper bound on the attractor dimension established with the use of a suitable anisotropic version of the Lieb–Thirring inequality. Estimates of the energy dissipation rate for boundary driven flows, essential in turbulence theory, are investigated, e.g., in [230] for ˝ D .0; Lx / .0; Ly / .0; h/, and in [229] for a smooth and bounded three-dimensional domains. The best estimates come from variational methods, cf. [88], and, in particular [134]. Other contexts and problems can be found, e.g., in monographs [62, 88, 99, 197, 220], and the literature quoted there. Boundary driven flows in smooth and bounded two-dimensional domains for a non-autonomous Navier–Stokes system are considered in [178], by using an approach of Chepyzhov and Vishik, cf. [62]. In Chap. 13 we shall consider a shear flow with time-dependent boundary driving.

10

Exponential Attractors in Contact Problems

In this chapter we consider two examples of contact problems. First, we study the problem of time asymptotics for a class of two-dimensional turbulent boundary driven flows subject to the Tresca friction law which naturally appears in lubrication theory. Then we analyze the problem with the generalized Tresca law, where the friction coefficient can depend on the tangential slip rate. While the first problem is governed by a variational inequality, since the underlying potential is convex, the problem with the generalized Tresca law due to the lack of potential convexity leads to a differential inclusion where the multivalued term has the form of the Clarke subdifferential. We prove that in both cases the global attractors exist and they have finite fractal dimensions. Moreover, there exists an object, called exponential attractor. It contains the global attractor, has finite fractal dimension and attracts the trajectories exponentially fast in time. This property allows, among other things, to locate the exponential and thus the global attractor in the phase space by using numerical analysis. We start the chapter with the section in which we remind a general method of proving the existence of exponential attractors.

10.1 Exponential Attractors and Fractal Dimension Let us consider an abstract autonomous evolutionary problem dv.t/ D F.v.t// in dt v.0/ D v0 ;

Z

for a.e.

t 2 RC ;


(10.1)

207

208

10 Exponential Attractors in Contact Problems

where Z is a Banach space, F is a nonlinear operator, and v0 is in a Banach space X that embeds in Z. We assume that the above problem has a global in time unique solution RC 3 t ! v.t/ 2 X for every v0 2 X. In this case one can associate with the problem a semigroup fS.t/gt0 of (nonlinear) operators S.t/ W X ! X setting S.t/v0 D v.t/, where v.t/, t > 0, is the unique solution of (10.1). Below we recall the notion of the global attractor and its properties (cf. Sect. 7.2). We know that from the properties of the semigroup of operators fS.t/gt0 we can derive the information on the behavior of solutions of problem (10.1), in particular, on their time asymptotics. One of the objects, the existence of which characterizes the asymptotic behavior of solutions, is the global attractor. It is a compact and invariant with respect to operators S.t/ subset of the phase space X (in general, a metric space) that uniformly attracts all bounded subsets of X. Definition 10.1. A global attractor for a semigroup fS.t/gt0 in a Banach space X is a subset A of X such that: • A is compact in X, • A is invariant, i.e., S.t/A D A for every t 0, • for every S " > 0 and every bounded set B in X there exists t0 D t0 .B; "/ such that tt0 S.t/B is a subset of the "-neighborhood of the attractor A (uniform attraction property). The global attractor defined above is uniquely determined by the semigroup. Moreover, it is connected and also has the following properties: it is the maximal compact invariant set and the minimal closed set that attracts all bounded sets. The global attractor may have a very complex structure. However, as a compact set (in an infinite dimensional Banach space) its interior is empty. In Sect. 7.2 we proved the following theorem that guarantees the global attractor existence (see Theorem 7.3 and Remark 7.4). Theorem 10.1. Let fS.t/gt0 be a semigroup of operators in a Banach space X such that: • for all t 0 the operators S.t/ W X ! X are continuous, • fS.t/gt0 is dissipative, i.e., there exists a bounded setSB0 X such that for every bounded set B X there exists t0 D t0 .B/ such that tt0 S.t/B B0 , • fS.t/gt0 is asymptotically compact, i.e., for every bounded set B X and all sequences tk ! 1 and yk 2 S.tk /B, the sequence fyk g is relatively compact in X. Then fS.t/gt0 has a global attractor A . In this chapter, however, we shall make use of another general theorem on the existence of the global attractor, cf. Theorem 10.2. As we have seen in Chap. 9, for some dissipative dynamical systems the global attractor has a finite fractal dimension. This fact has a number of important consequences for the behavior of the flow generated by the semigroup [197, 198].

10.1 Exponential Attractors and Fractal Dimension

209

Definition 10.2. The fractal dimension (also known as upper box counting dimension) of a compact set K in a Banach space X is defined as dfX .K/ D lim sup "!0

log N"X .K/ ; log 1"

where N"X .K/ is the minimal number of balls of radius " in X needed to cover K. Another important property that holds for many dynamical systems is the existence of an exponential attractor. Definition 10.3. An exponential attractor for a semigroup fS.t/gt0 in a Banach space X is a subset M of X such that: • • • •

M is compact in X, M is positively invariant, i.e., S.t/M M for every t 0, fractal dimension of M is finite, i.e., dfX .M / < 1, M attracts exponentially all bounded subsets of X, i.e., there exist a universal constant c1 and a monotone function ˚ such that for every bounded set B in X, its image S.t/B is a subset of the ".t/-neighborhood of M for all t t0 , where ".t/ D ˚.kBkX /ec1 t (exponential attraction property).

Since the global attractor A is the minimal compact attracting set it follows that if both global and exponential attractors exist, then A M and the fractal dimension of A must be finite. Moreover, in contrast to the global attractor, an exponential attractor does not have to be unique. In the following we will remind the abstract framework of [168] (see also [162]) that uses the so-called method of l-trajectories (or short trajectories) to prove the existence of global and exponential attractors. Let X, Y, and Z be three Banach spaces such that Y X

with compact imbedding

and X Z:

We assume, moreover, that X is reflexive and separable. For > 0, let X D L2 .0; I X/; and du 2 L2 .0; I Z/ : Y D u 2 L2 .0; I Y/ W dt By Cw .Œ0; I X/ we denote the space of weakly continuous functions from the interval Œ0; to the Banach space X, and we assume that the solutions of (10.1) are at least in Cw .Œ0; TI X/ for all T > 0. Then by an l-trajectory we mean the restriction of any solution to the time interval Œ0; l. If v D v.t/; t 0, is the solution of (10.1)

210


then both D vjŒ0;l and all shifts Lt ./ D vjŒt;lCt for t > 0 are l-trajectories. Note that the mapping Lt is defined as Lt ./. / D v. C t/ for t > 0 and 2 Œ0; l, where v is a unique solution such that vjŒ0;l D . We can now formulate a theorem which gives criteria for the existence of a global attractor A for the semigroup fS.t/gt0 in X and its finite dimensionality. The following criteria are stated as assumptions .A1/–.A8/ in [168]: .A1/ for any v0 2 X and arbitrary T > 0 there exists (not necessarily unique) v 2 Cw .Œ0; TI X/ \ YT , a solution of the evolutionary problem on Œ0; T with v.0/ D v0 ; moreover, for any solution v, the estimates of kvkYT are uniform with respect to kv0 kX , .A2/ there exists a bounded set B0 X with the following properties: if v is an arbitrary solution with initial condition v0 2 X then 1. there exists t0 D t0 .kv0 kX / such that v.t/ 2 B0 for all t t0 , 2. if v0 2 B0 then v.t/ 2 B0 for all t 0, .A3/ each l-trajectory has a unique continuation among all solutions of the evolution problem which means that from an endpoint of an l-trajectory there starts at most one solution, .A4/ for all t > 0, Lt W Xl ! Xl is continuous on Bl0 —the set of all l-trajectories starting at any point of B0 from .A2/, .A5/ for some > 0, the closure in Xl of the set L .Bl0 / is contained in Bl0 , .A6/ there exists a space Wl such that Wl Xl with compact embedding, and > 0 such that L W Xl ! Wl is Lipschitz continuous on Bl1 —the closure of L .Bl0 / in Xl , .A7/ the map e W Xl ! X, e./ D .l/, is continuous on Bl1 , .A8/ the map e W Xl ! X is Hölder-continuous on Bl1 . Theorem 10.2. Let the assumptions .A1/–.A5/, .A7/ hold. Then there exists a global attractor A for the semigroup fS.t/gt0 in X. Moreover, if the assumptions .A6/, .A8/ are satisfied then the fractal dimension of the attractor is finite. For the existence of an exponential attractor we need two additional properties, where now X is a Hilbert space (cf. [168]): .A9/ for all > 0 the operators Lt W Xl ! Xl are (uniformly with respect to t 2 Œ0; ) Lipschitz continuous on Bl1 , .A10/ for all > 0 there exists c > 0 and ˇ 2 .0; 1 such that for all 2 Bl1 and t1 ; t2 2 Œ0; it holds that kLt1 Lt2 kXl cjt1 t2 jˇ :

(10.2)

Theorem 10.3. Let X be a separable Hilbert space and let the assumptions .A1/– .A6/ and .A8/–.A10/ hold. Then there exists an exponential attractor M for the semigroup fS.t/gt0 in X. For the proofs of Theorems 10.2 and 10.3 we refer the readers to corresponding theorems in [168].

10.2 Planar Shear Flows with the Tresca Friction Condition

211

10.2 Planar Shear Flows with the Tresca Friction Condition 10.2.1 Problem Formulation We start this section from the description of the problem for which we will later prove the exponential attractor existence. The flow of an incompressible fluid in a two-dimensional domain ˝ is described by the equation of motion ut u C .u r/u C rp D 0

in ˝ RC ;

(10.3)

and the incompressibility condition in ˝ RC :

div u D 0

(10.4)

To define the domain ˝ of the flow, let ˝1 be the channel ˝1 D fx D .x1 ; x2 / W 1 < x1 < 1; 0 < x2 < h.x1 /g; where h is a positive function, smooth, and L-periodic in x1 . Then we set ˝ D fx D .x1 ; x2 / W 0 < x1 < L; 0 < x2 < h.x1 /g; and @˝ D C [ L [ D , where C and D are the bottom and the top, and L is the lateral part of the boundary of ˝. By n we will denote the unit outward normal to @˝ and by T D T.u; p/ the stress tensor given by T.u; p/ D .Tij .u; p//2i;jD1 , where Tij .u; p/ D pıij C ui;j C uj;i . We are interested in solutions of (10.3)–(10.4), such that the velocity u and the Cauchy stress vector T.u; p/n on L are L-periodic with respect to x1 . Note that if both velocity and pressure are smooth functions defined on ˝1 and L-periodic with respect to x1 , then the periodicity of the Cauchy stress vector on L follows automatically from the constitutive law. We assume that uD0

on

D RC :

(10.5)

Moreover, we assume the no flux condition across C so that the normal component of the velocity on C satisfies un D u n D 0

on

C RC ;

(10.6)

and that the tangential component of the velocity u on C is unknown and satisfies the Tresca friction law with a constant and positive maximal friction coefficient k. This means that, cf., e.g., [91, 215],

212


9 > > > > > =

jT .u; p/j k jT .u; p/j < k ) u D U0 e1 jT .u; p/j D k ) 9 0 such that u D U0 e1 T .u; p/

> > > > > ;

on

C RC ;

(10.7) where U0 e1 D .U0 ; 0/, U0 2 R, is the velocity of the lower surface producing the driving force of the flow and T is the tangential component of the stress vector on C given by T .u; p/ D T.u; p/n ..T.u; p/n/ n/n;

(10.8)

and u is the tangential velocity on C given by u D u un n. From (10.6) it follows that u D u. Finally, the initial condition for the velocity field is u.x; 0/ D u0 .x/

for

x 2 ˝:

The problem is motivated by the examination of a certain two-dimensional flow in an infinite (rectified) journal bearing ˝ .1; C1/, where D .1; C1/ represents the outer cylinder, and C .1; C1/ represents the inner, rotating cylinder. In the lubrication problems the gap h between cylinders is never constant. We can assume that the rectification does not change the equations as the gap between cylinders is very small with respect to their radii. Since the widely used no-slip boundary condition when the fluid has the same velocity as surrounding solid boundary is not respected if the shear rate becomes too high we can model such situation by a transposition of the well-known friction laws between two solids [215] to the fluid–solid interface. The Tresca friction law is one of them. To work with the above problem it is convenient to transform the boundary condition (10.7) to the homogeneous one. To this end, let u.x1 ; x2 ; t/ D U.x2 /e1 C v.x1 ; x2 ; t/;

(10.9)

with U.0/ D U0 ;

U.h.x1 // D 0;

x1 2 .0; L/:

(10.10)

The new vector field v is L-periodic in x1 and satisfies the equation of motion vt v C .v r/v C rp D G.v/; with G.v/ D Uv;x1 .v/2 U;x2 e1 C U;x2 x2 e1 ;

(10.11)


213

where by .v/2 we denoted the second component of v. As div.Ue1 / D 0 we get div v D 0

in ˝ RC :

(10.12)

D RC

(10.13)

From (10.9)–(10.10) we obtain vD0

on

and vn D v n D 0

on

C RC :

(10.14)

Moreover, T .v; p/ D T .u; p/ C

! ˇ @U.x2 / ˇˇ ;0 @x2 ˇx2 D0

on C RC :

Since we can define the extension U in such a way that ˇ @U.x2 / ˇˇ D 0; @x2 ˇx2 D0 the Tresca condition (10.7) transforms to 9 > > > > > =

jT .v; p/j k jT .v; p/j < k ) v D 0 jT .v; p/j D k ) 9 0 such that v D T .v; p/

> > > > > ;

on

C RC :

(10.15) Finally, the initial condition becomes v.x; 0/ D v0 .x/ D u0 .x/ U.x2 /e1

for x 2 ˝:

(10.16)

The Tresca condition (10.15) is a particular case of an important in contact mechanics class of subdifferential boundary conditions of the form, cf., e.g., [188], '. .x//'.v .x; t// T .x; t/. .x/v .x; t//

on

C RC ; (10.17)

where belongs to a certain set of admissible functions (in our case the tangential components of the traces on C of functions from the space V defined below) and ' is a convex functional. For '.v / D kjv j the last condition is equivalent to (10.15).

214


We pass to the variational formulation of the homogeneous problem (10.11)– (10.16). Then, for the convenience of the reader, we describe the relations between the classical and the weak formulations. We begin with some basic definitions. Let VQ D fv 2 C1 .˝/2 W div v D 0 in ˝; v D 0 on D ;

v is L-periodic in x1 ;

v n D 0 on C g;

and V D closure of VQ in H 1 .˝/2 ;

H D closure of VQ in L2 .˝/2 :

We define the scalar products in H and V, respectively, by Z

Z .u ; v/ D

˝

u.x/ v.x/ dx

and

.ru ; rv/ D

ru.x/W rv.x/ dx; ˝

and their associated norms by 1

jvj D .v; v/ 2

1

and

kvk D .rv ; rv/ 2 :

and

b.u ; v ; w/ D ..u r/v ; w/:

Let, for u; v, and w in V, a.u ; v/ D .ru ; rv/

We denote the trace operator from V to L2 .C /2 by . Its norm is denoted by k k D k kL .VIL2 .C /2 / . Finally, let us define the convex and continuous functional ˚ W V ! R by Z

Z ˚.u/ D

C

kju.x1 ; 0/jdx1 D

C

'.u / dS:

The variational formulation of the homogeneous problem (10.11)–(10.16) is as follows: Problem 10.1. Given v0 2 H, find v W Œ0; 1/ ! H such that: .i/ for all T > 0, v 2 C.Œ0; TI H/ \ L2 .0; TI V/ where V 0 is the dual space to V,

with vt 2 L2 .0; TI V 0 /;


215

.ii/ for all in V, and for almost all t in the interval .0; 1/, the following variational inequality holds hvt .t/; v.t/i C a.v.t/; v.t// C b.v.t/; v.t/; v.t// C ˚./ ˚.v.t// hL .v.t//; v.t/i; (10.18) .iii/ the following initial condition holds v.x; 0/ D v0 .x/

for a.e.

x 2 ˝:

(10.19)

In (10.18) the operator L W V ! V 0 is defined by hL .v/; i D a.; / b.; v; / b.v; ; /; where D Ue1 is a suitable smooth background flow. We have the following relations between the classical and weak formulations. Proposition 10.1. Every classical solution of Problem (10.11)–(10.16) is also a solution of Problem 10.1. On the other hand, every solution of Problem 10.1 which is smooth enough is also a classical solution of Problem (10.11)–(10.16). Proof. As the proof is quite technical it is only sketched, and the details are left to the reader. Let v be a classical solution of Problem (10.11)–(10.16). As it is (by assumption) sufficiently regular, we have to check only (10.18). Remark first that (10.11) can be written as vt .t/ div T.v.t/; p.t// C .v.t/ r/v.t/ D G.v.t//:

(10.20)

Let 2 V. Multiplying (10.20) by v.t/ and using the Green formula we obtain Z

Z ˝

vt .t/ . v.t// dx C Z

D

@˝

˝

Tij .v.t/; p.t//. v.t//i;j dx C b.v.t/; v.t/; v.t// Z

Tij .v.t/; p.t//nj . v.t//i dS C

˝

G.v.t// . v.t// dx

for t 2 .0; T/. As v.t/ and are in V, after some calculations we obtain Z Tij .v.t/; p.t//. v.t//i;j dx D a.v.t/ ; v.t//:

(10.21)

(10.22)

˝

By (10.17) with '.v / D kjv j, taking into account the boundary conditions, we get Z Z Tij .v.t/; p.t//nj . v.t//i dS D T .v.t/; p.t// . v .t// dS @˝

C

Z

C

k.j j jv .t/j/ dS:

(10.23)

216


Finally, Z ˝

G.v.t// . v.t//dx D hL .v.t// ; v.t/i:

(10.24)

From (10.22)–(10.24) we see that (10.21) yields (10.18), and (10.19) is the same as (10.16). Conversely, suppose that v is a sufficiently smooth solution to Problem 10.1. We have immediately (10.12)–(10.14) and (10.16). Let ' be in the space 1 .Hdiv .˝//2 D f' 2 V W ' D 0 on @˝g:

We take D v.t/ ˙ ' in (10.18) to get hvt .t/ v.t/ C .v.t/ r/v.t/ G.v.t// ; 'i D 0

for every

1 ' 2 .Hdiv .˝//2 :

Thus, there exists a distribution p.t/ on ˝ such that vt .t/ v.t/ C .v.t/ r/v.t/ G.v.t// D rp.t/

in

˝;

(10.25)

so that (10.11) holds. It follows that p is smooth and without loss of generality we may assume that it is also L-periodic with respect to x1 . Now, we shall derive the Tresca boundary condition (10.15) from the weak formulation. We have Z

Z ˝

Tij .v.t/; p.t//. v.t//i;j dx D Z

˝

C @˝

div T.v.t/; p.t// . v.t// dx

T.v.t/; p.t//n . v.t// dS: (10.26)

Applying (10.22) and (10.26) to (10.18) we get Z ˝

.vt .t/ div T.v.t/; p.t// C .v.t/ r/v.t/ G.v.t/// . v.t// dx Z C @˝

T.v.t/; p.t//n . v.t// dS ˚.v.t// ˚./:

By (10.25) we have (10.20) and so the first integral on the left-hand side vanishes. Thus we obtain condition (10.23). Due to the fact that n vn .t/ D 0 on C , we conclude Z T.v.t/; p.t//n . v.t//i dS @˝

Z D C

Z T .v.t/; p.t// . v .t// dS C

L

T.v.t/; p.t//n . v.t// dS;


and then, Z

217

Z

L

T.v.t/; p.t//n . v.t// dS C Z

C

T .v.t/; p.t// . v .t// dS

C

k.j j jv .t/j/ dS;

where is any element of V. We assume that the weak solution v is a sufficiently smooth and L-periodic with respect to x1 function defined on the infinite strip ˝1 . Then, as the associated pressure is also L-periodic, it follows that the Cauchy stress vector T.v.t/; p.t//n must be L-periodic with respect to x1 and the integral on L vanishes, whereas Z Z T .v.t/; p.t// . v .t// dS k.j j jv .t/j/ dS: (10.27) C

C

Taking D 0 and D 2v, respectively, we get Z

Z

C

T v dS D

whence it follows that Z Z T dS C

C

kj j dS

C

kjv j dS;

for every

2 V:

(10.28)

(10.29)

Note that only the second coordinate of T and v is nonzero on C , so by a slight abuse of notation we can consider these functions as scalars. If is smooth on C , using the methods of Sect. 1.2 in [146] it is possible to construct 2 V such that its tangential part is equal to on C . From density of smooth functions in L2 .C / it follows that the following inequality analogous to (10.29) Z

Z C

T dS

kj j dS C

holds for all 2 L2 .C /. From the above inequality it follows that the pointwise inequality jT j k must hold a.e. on C . Indeed, if the opposite inequality holds on a set S of positive measure then we arrive at contradiction by taking D 0 outside S and D jTT j on S. Now, (10.28) implies that the Tresca condition (10.15) holds and the proof is complete. t u

218


10.2.2 Existence and Uniqueness of a Global in Time Solution In this subsection we establish the existence and uniqueness of a global in time solution for Problem 10.1. First, we present two lemmas. Lemma 10.1 (cf. [28]). There exists a smooth extension .x1 ; x2 / D U.x2 /e1 of U0 e1 from C to ˝ satisfying: (10.10), ˇ @U.x2 / ˇˇ D 0; @x2 ˇx2 D0 and such that jb.v ; ; v/j

kvk2 4

v 2 V:

for all

Moreover, jj2 C jrj2 D

Z ˝

jU.x2 /j2 dx1 dx2 C

Z ˝

jU;x2 .x2 /j2 dx1 dx2 F;

where F is a constant depending on ; ˝, and U0 . Exercise 10.1. Provide a proof of Lemma 10.1, cf. Lemma 9.2. Lemma 10.2. For all v in V we have the Ladyzhenskaya inequality 1

1

kvkL4 .˝/ C.˝/jvj 2 kvk 2 :

(10.30)

Proof. Let v 2 V and r 2 C1 .ŒL; L/ such that r D 1 on Œ0 ; L and r D 0 at x1 D L. Define ' D rv (keep in mind that v can be periodically extended to whole ˝1 ), and extend ' by 0 to ˝1 D .L ; L/ .0 ; h/, where h D max0x1 L h.x1 /. We obtain ˇ ˇ Z x1 Z L ˇ @' ˇ @' 2 ˇ ' .x1 ; x2 / D 2 '.t1 ; x2 / .t1 ; x2 / dt1 2 j'.x1 ; x2 /j ˇ .x1 ; x2 /ˇˇ dx1 @t1 @x1 L L and ' 2 .x1 ; x2 / D 2

Z

h x2

'.x1 ; t2 /

@' .x1 ; t2 / dt2 2 @t2

Z

h 0

ˇ ˇ ˇ @' ˇ j'.x1 ; x2 /j ˇˇ .x1 ; x2 /ˇˇ dx2 ; @x 2


219

whence k'k4L4 .˝1 / D

Z ˝1

Z

' 2 .x1 ; x2 /' 2 .x1 ; x2 / dx1 dx2 h

sup

0 Lx1 L

Z 4

h

Z

0

! Z

2

' .x1 ; x2 / dx2

!

L

2

sup ' .x1 ; x2 / dx1

L 0x2 h

ˇ ˇ ˇ ˇ Z L Z h ˇ @' ˇ ˇ @' ˇ ˇ ˇ ˇ ˇ dx1 dx2 dx2 dx1 : j'j ˇ j'j ˇ @x1 ˇ @x2 ˇ L L 0 L

By the Cauchy–Schwartz inequality, k'k4L4 .˝1 /

4j'j2L2 .˝1 /

2j'j2L2 .˝1 /

ˇ ˇ ˇ @' ˇ ˇ ˇ ˇ @x ˇ

1 L2 .˝1 /

ˇ ˇ ˇ @' ˇ2 ˇ ˇ ˇ @x ˇ 2

ˇ ˇ ˇ @' ˇ ˇ ˇ ˇ @x ˇ

2 L2 .˝1 /

1 L .˝1 /

ˇ ˇ ˇ @' ˇ2 ˇ ˇ Cˇ @x ˇ 2

!

2 L .˝1 /

2j'j2L2 .˝1 / jr'j2L2 .˝1 / : We use the fact that jrj 1 and the Poincaré inequality to get kvkL4 .˝/ k'kL4 .˝1 / ;

j'jL2 .˝1 / 2jvj;

and

jr'jL2 .˝1 / Ckvk t u

for some constant C, whence (10.30) holds.

Theorem 10.4. For any v0 2 H and U0 2 R there exists a solution of Problem 10.1. Proof. We provide only the main steps of the proof as it is quite standard and, on the other hand, long. The estimates we obtain will be used further in the section. Observe that the functional ˚ is convex, lower semicontinuous (in fact it is continuous) but nondifferentiable at zero. To overcome this difficulty we use the following approach (see, e.g., [91, 113, 188]). For ı > 0 let ˚ı W V ! R be a functional defined by Z 1 V 3 w 7! ˚ı .w/ D kjw j1Cı dx: 1 C ı C This functional is convex and lower semicontinuous on V. Moreover, for vı ! v weakly in L2 .0; TI V/, Z T Z T ˚ı .vı .t// dt ˚.v.t// dt; lim inf ı!0C

0

0

and lim ˚ı .w/ D ˚.w/

ı!0C

220


for all w 2 V. The functional ˚ı is Gâteaux differentiable in V, with Z 0 h˚ı .v/ ; i D kjv jı1 v for all 2 V: C

Let us consider the following equation,

dvı .t/ ; C a.vı .t/; / C b.vı .t/; vı .t/; / C h˚ı0 .vı .t//; i dt D a.; / b.; vı .t/; / b.vı .t/; ; /

(10.31)

for any test function 2 V, with the initial condition vı .0/ D v0 :

(10.32)

For ı > 0, we establish a priori estimates of vı . Since h˚ı0 .vı /; vı i 0, vı 2 V, and b.vı ; vı ; vı / D b.; vı ; vı / D 0, taking D vı .t/ in (10.31) we get 1d jvı .t/j2 C kvı .t/k2 a.; vı .t// b.vı .t/; ; vı .t//: 2 dt In view of Lemma 10.1 we obtain 1d jvı .t/j2 C kvı .t/k2 kk2 : 2 dt 2 We estimate the right-hand side in terms of the data using Lemma 10.1 to get 1d jvı .t/j2 C kvı .t/k2 F; 2 dt 2

(10.33)

with F D F.; ˝; U0 /. From (10.33) we conclude that jvı .t/j2 C

Z 0

t

kvı .s/k2 ds jv.0/j2 C 2tF;

(10.34)

whence vı is bounded in L2 .0; TI V/ \ L1 .0; TI H/ independently of ı:

(10.35)

The existence of vı satisfying (10.31)–(10.32) is based on inequality (10.33), the Galerkin approximations, and the compactness method. Moreover, from (10.34) we can deduce that dvı dt

is bounded in

L2 .0; TI V 0 /:

(10.36)


221

From (10.35) and (10.36) we conclude that there exists v such that (possibly for a subsequence) vı ! v weakly in L2 .0; TI V/;

dv dvı ! weakly in L2 .0; TI V 0 /: (10.37) dt dt

In view of (10.37), v 2 C.Œ0; TI H/, and vı ! v

strongly in L2 .0; TI H/:

We can now pass to the limit ı ! 0 in (10.31)–(10.32) as in [91] to obtain the variational inequality (10.18) for almost every t 2 .0; T/. Thus the existence of a solution of Problem 10.1 is established. t u Theorem 10.5. Under the hypotheses of Theorem 10.4, the solution v of Problem 10.1 is unique and the map v. / ! v.t/, for t > 0, is Lipschitz continuous in H. Proof. Let v and w be two solutions of Problem 10.1. Set D w in the variational inequality for v, D v in the variational inequality for w, and add the two thus obtained inequalities. The terms with the boundary functionals reduce and for u.t/ D w.t/ v.t/ we obtain 1d ju.t/j2 C ku.t/k2 b.u.t/; w.t/; u.t// C b.u.t/; ; u.t//: 2 dt By Lemma 10.1 and the Ladyzhenskaya inequality (10.30) we obtain d 2 ju.t/j2 C ku.t/k2 C.˝/4 kw.t/k2 ju.t/j2 ; dt 2

(10.38)

and then in view of the Poincaré inequality we conclude d 2 ju.t/j2 C ju.t/j2 C.˝/4 kw.t/k2 ju.t/j2 ; dt 2 where > 0 is a constant. Using the Gronwall inequality, we obtain Z t 2 C.˝/4 kw.s/k2 ds : ju.t/j2 ju. /j2 exp 2

(10.39)

From (10.37) it follows that the solution w of Problem 10.1 belongs to L2 .; tI V/. By (10.39) the map v. / ! v.t/, t > 0, in H is Lipschitz continuous, with jw.t/ v.t/j Cjw. / v./j

(10.40)

uniformly for t; in a given interval Œ0; T and initial conditions w.0/; v.0/ in a given bounded set B in H.

222


In particular, as u.0/ D w.0/ v.0/ D 0, the solution v of Problem 10.1 is unique. This ends the proof of Theorem 10.5. u t

10.2.3 Existence of Finite Dimensional Global Attractor In this subsection we prove the following theorem: Theorem 10.6. There exists a global attractor of a finite fractal dimension for the semigroup fS.t/gt0 associated with Problem 10.1. Proof. From the considerations in the previous section it follows that to prove the theorem it suffices to check assumptions .A1/–.A6/, .A8/. For the convenience of the reader we repeat their statements in appropriate places. Assumption .A1/. For any v0 2 X and arbitrary T > 0 there exists (not necessarily unique) v 2 Cw .Œ0; TI X/ \ YT , a solution of the evolutionary problem on Œ0; T with v.0/ D v0 . Moreover, for any solution the estimates of kvkYT are uniform with respect to kv.0/kX . In our case, set X D H, Y D V, and YT D fu 2 L2 .0; TI V/ W u0 2 L2 .0; TI V 0 /g. From Theorems 10.4 and 10.5 we know that for any v0 2 H and arbitrary T > 0 there exists a unique v 2 C.Œ0; TI H/ \ YT , solution of Problem 10.1. We shall obtain the needed estimates directly from the variational inequality, cf. (10.18), hvt .t/; v.t/i C a.v.t/; v.t// C b.v.t/; v.t/; v.t// C ˚./ ˚.v.t// hL .v.t//; v.t/i:

(10.41)

Set D 0 in (10.41) to get 1d 2 jvj C kvk2 C ˚.v/ hL .v/; vi 2 dt as ˚.0/ D 0. Since, by Lemma 10.1, hL .v/; vi

kvk2 C kk2 2

we obtain d 2 jvj C kvk2 C 2˚.v/ 2kk2 D F: dt

(10.42)


223

Integration in t gives jv.t/j2 C

Z

t 0

kv.s/k2 ds jv.0/j2 C 2tF;

(10.43)

and we deduce that is bounded in L2 .0; TI V/ \ L1 .0; TI H/

v

uniformly with respect to jv.0/j. To get a uniform with respect to jv.0/j estimate of v 0 in L2 .0; TI V 0 / set D v , 2 V, in (10.41). We then have hv 0 ; i hL .v/; i a.v; / b.v; v; / C ˚.v

/ ˚.v/:

(10.44)

Using the Poincaré and the trace inequalities we obtain k.v/kL2 .@˝/ Ckvk, and Z

Z ˚.v

/ ˚.v/ D k

C

.jv

j

jv j/ dS k

j C

j dS

C.C /k k: (10.45)

Moreover, using the Ladyzhenskaya inequality (10.30) to the nonlinear term, we have hL .v/; i a.v; / b.v; v; / C1 .kvk C jvj kvk C 1/k k:

(10.46)

From (10.44)–(10.46) we obtain kv 0 kV 0 C2 .kvk C jvj kvk C 1/; and kv 0 k2L2 .0;TIV 0 / D

Z

T

0

kv 0 .t/k2V 0 dt

Z

C2

T 0

kv.t/k2 dtCkvk2L1 .0;TIH/

Z

T

kv.t/k2 dtCT C.jv.0/j/:

0

Thus, .A1/ holds true. Assumption .A2/. There exists a bounded set B0 X with the following properties: if v is an arbitrary solution with initial condition v0 2 X then (i) there exists t0 D t0 .kv0 kX / such that v.t/ 2 B0 for all t t0 , (ii) if v0 2 B0 then v.t/ 2 B0 for all t 0. From (10.42) and the Poincaré inequality, d 2 jvj C 1 jvj2 F; dt

224


and then by the Gronwall inequality (cf. Lemma 3.20) jv.t/j2 jv.0/j2 e1 t C

F : 1

Thus, there exists a bounded absorbing set (e.g., the ball BH .0; / with 2 D 2 F 1 ) S H in H. Let B0 be defined as B0 D t0 S.t/BH .0; / . If v0 2 B0 then there exist sequences tk and vk ! v0 in H such that vk 2 S.tk /BH .0; /. From the continuity of S.t/ it follows that S.t C tk /BH .0; / 3 S.t/vk ! S.t/v0 in H and hence S.t/v0 2 B0 for all t 0. Thus, .A2/ holds true. Note that we need B0 to be closed in H to be able to satisfy assumption .A5/ below. Assumption .A3/. Each l-trajectory has among all solutions a unique continuation. We recall that by the l-trajectory we mean any solution on the time interval Œ0; l, and the unique continuation means that from an endpoint of an l-trajectory there starts at most one solution. In our case .A3/ is satisfied as the solutions are unique (Theorem 10.5). Assumption .A4/. For all t > 0, Lt W Xl ! Xl is continuous on Bl0 . Bl0 is defined as the set of all l-trajectories starting at any point of B0 from .A2/, and Xl D L2 .0; lI X/. The semigroup fLt gt0 acts on the set of l-trajectories as the shifts operators: .Lt /. / D v.t C / for 0 l, where v is the unique solution on Œ0; l C such that vjŒ0;l D . In our case, X D H, hence Xl D Hl D L2 .0; lI H/. In view of inequality (10.40) the map S.t/ W B0 ! B0 is Lipschitz continuous for every t > 0, whence for any two 1 ; 2 in Bl0 , Z

l 0

2

2

jLt 1 .s/ Lt 2 .s/j ds C .t/

Z

l 0

j1 .s/ 2 .s/j2 ds:

(10.47)

Thus, .A4/ holds true. Assumption .A5/. For some > 0, the closure in Xl of the set L .Bl0 / is included in Bl0 . As L .Bl0 / Bl0 , it is enough to check that the set Bl0 is closed in Xl . We have to prove that if fk g is a sequence in Bl0 converging to some in Xl then is also a trajectory and that .0/ 2 B0 . From assumption .A1/ it follows that the sequence fk g is bounded in Yl and thus contains a subsequence (relabeled fk g) such that k ! weakly in L2 .0; lI V/ and

d dk ! weakly in L2 .0; lI V 0 /: (10.48) dt dt

Moreover, by the Aubin–Lions compactness theorem, k !

strongly in L2 .0; lI H/:

(10.49)


225

We have h0k .t/; k .t/i C a.k .t/; k .t// C b.k .t/; k .t/; k .t// C ˚./ ˚.k .t// hL .k .t// ; k .t/i: We multiply both sides by a nonnegative smooth function D .t/ with support in the interval .0; l/ and integrate with respect to t in this interval. We shall prove that taking lim infk!1 of both sides and using (10.48) and (10.49) we obtain Z

l 0

h0 .t/; .t/i.t/ dt C Z

l

C 0

Z

l 0

a..t/; .t//.t/dt Z

b..t/; .t/; .t//.t/ dt C

Z

l

0

l 0

Z ˚./.t/ dt

0

l

˚..t//.t/ dt

hL ..t//; .t/i.t/ dt:

(10.50)

First we shall show (without using the convexity argument) that Z lim

l

n!1 0

Z ˚.k .t//.t/dt D

0

l

˚..t//.t/dt:

(10.51)

Let 2 0; 12 . From the fact that H 1 .˝/ embeds in H 1 .˝/ compactly, the trace operator 1 W H 1 .˝/ ! L2 .@˝/ is linear and continuous, and from the Ehrling lemma (cf. Lemma 3.17) for the spaces H 1 .˝/; H 1 .˝/, and L2 .˝/ it follows that for every " > 0 there exists C" > 0 such that for all v 2 H 1 .˝/, k.v/kL2 .@˝/ "jrvj C C" jvj: We have thus k.k / ./k2L2 .C / "kk k2 C C"0 jk j2 ; and Z 0

l

k.k / ./k2L2 .C / dt "

Z

l 0

kk k2 dt C C"0

Z

As, in view of (10.48), there exists M > 0 such that for all n, Z 0

l

kk .t/ .t/k2 dt M;

0

l

jk j2 dt:

226


we obtain, using (10.49), Z

l

lim sup 0

k!1

k.k / ./k2L2 .C / dt "M:

Now, as " is any positive number, we obtain Z

l

lim

k!1 0

k.k / ./k2L2 .C / dt D 0:

(10.52)

From (10.52), (10.51) easily follows. We have also Z lim

k!1 0

l

h0k ; k i.t/ dt

Z D lim

l

k!1 0

D lim

Z

1d jk j2 .t/ dt 2 dt

k!1 0

Z

l

D 0

l

1 jk j2 .t/0 dt D 2

Z

l 0

1 2 jj .t/0 dt 2

h0 ; i.t/ dt:

In view of (10.48) and (10.49), there are no problems to get the other terms in (10.50) and finally, (10.50) itself. As inequality (10.50) is equivalent to the inequality h.t/0 ; i C a..t/; .t// C b..t/; .t/; .t// C ˚./ ˚..t// hL ..t//; .t/i; satisfied for almost all t 2 .0; l/, is a solution with .0/ in H. To end the proof we have to show that .0/ belongs to the set B0 . We have k .t/ 2 B0 for all t 2 Œ0; l and, by (10.49), for a subsequence, k .t/ ! .t/ for almost all t 2 .0; l/. As B0 is closed, .t/ 2 B0 for almost all t 2 .0; l/. Now, from the continuity of W Œ0; l ! H and the closedness of B0 it follows that .0/ is in B0 . Thus, assumption .A5/ holds. Assumption .A6/. There exists a space Wl such that Wl Xl with compact embedding, and > 0 such that L W Xl ! Wl is Lipschitz continuous on Bl1 —the closure in Xl of L .Bl0 /. Define Wl D fu 2 L2 .0; lI V/ W u0 2 L1 .0; lI U 0 /g; where U D f 2 V W D 0 on C g. We have, Wl Xl , with compact embedding and we shall prove that Ll W Xl ! Wl is Lipschitz continuous on Bl1 . Let w and v be two solutions of Problem 10.1 starting from B0 and let u D w v. Then, cf. (10.38),


227

d 2 ju.t/j2 C ku.t/k2 C.˝/4 kw.t/k2 ju.t/j2 : dt 2 Take s 2 .0; l/ and integrate this inequality over 2 .s; 2l/ to get Z 2l Z 2l 2 ju.2l/j2 C ku. /k2 d C.˝/4 ku. /k2 ju. /j2 d C ju.s/j2 : 2 s s (10.53) From (10.40) we conclude that ju. /j2 C1 ju.s/j2 for 2 .s; 2l/ and from (10.43) we have Z 2l Z 2l 1 ku. /k2 d .kv./k2 C kw. /k2 / d .jv.0/j2 C jw.0/j2 C 8lF/; s 0 whence Z

2l

ku. /k2 d C2

s

uniformly for w.s/; v.s/ 2 B0 . Therefore, from (10.53) we obtain Z 2l ku. /k2 d C3 ju.s/j2 : l

Integrating over s 2 .0; l/ we obtain Z l

2l

C3 ku. /k d l 2

Z

l

ju.s/j2 ds;

0

and therefore r kLl 1 Ll 2 kL2 .0;lIV/

C3 k1 2 kL2 .0;lIH/ l

(10.54)

for any 1 ; 2 in Bl0 . In order to prove that k.Ll 1 Ll 2 /0 kL1 .0;lIU0 / Ck1 2 kL2 .0;lIH/ for some C > 0, it is sufficient, in view of (10.54), to prove that k.1 2 /0 kL1 .0;lIU0 / C0 k1 2 kL2 .0;lIV/ with some C0 > 0.

(10.55)

228


We have hv 0 ; vi C a.v; v/ C b.v; v; v/ C ˚./ ˚.v/ hL .v/; vi; and hw0 ; wi C a.w; w/ C b.w; w; w/ C ˚./ ˚.w/ hL .w/; wi:

Setting D v in the first inequality and D w C in the second one, where 2 U, k k 1, and adding the obtained two inequalities we get hu0 ; i N .u; w; vI /;

(10.56)

where N .u; w; vI / D b.; u; / C b.u; ; / C a.u; / C b.w; u; / C b.u; v; /: Estimating the right-hand side of (10.56) we get hu0 ; i C00 .1 C kvk C kwk/kuk k k; whence ku0 .t/kU0 D supfhu0 .t/; i W k k 1g C00 .1 C kv.t/k C kw.t/k/ku.t/k: Finally, integration over t 2 .0; l/ gives Z 0

l

ku0 .t/kU0 dt C000

Z

l

ku.t/k2 dt

1=2

0

with C000 D C0

Z

l 0

.1 C kv.t/k2 C kw.t/k2 / dt

1=2 with a constant

C0 > 0;

uniformly for trajectories starting from B0 . This proves (10.55) and ends the proof of the Lipschitz continuity of the map Ll W Xl ! Wl . Assumption .A6/ holds true. Assumption .A8/. The map e W Xl ! X is Hölder-continuous on Bl1 . Assumption .A8/ follows directly from the Lipschitz continuity of the map e W Xl ! X, e./ D .l/. To check the latter, let w; v be two solutions as above, starting from B0 . From (10.40) we have, in particular, jw.l/ v.l/j Cjw. / v./j


229

for 2 .0; l/. Integrating this inequality in on the interval .0; l/ we obtain C jw.l/ v.l/j p kw vkL2 .0;lIH/ : l t u

This ends the proof of Theorem 10.6.

10.2.4 Existence of an Exponential Attractor Now we can prove the main theorem of Sect. 10.2. Theorem 10.7. There exists an exponential attractor for the semigroup fS.t/gt0 associated with Problem 10.1. Proof. In view of Theorem 10.3 and the considerations in the previous section it suffices to check conditions .A9/ and .A10/. The first one follows immediately from inequality (10.47). Thus it remains to prove that kLt1 Lt2 kXl cjt1 t2 jˇ

(10.57)

holds for all > 0, 0 t2 t1 , 2 Bl1 , some c > 0 and some ˇ 2 .0; 1, where, in our case, Xl D Hl . To obtain (10.57) it suffices to know that 0 , the time derivatives of 2 Bl1 , are uniformly bounded in Lq .0; lI H/ for some 1 < q 1, cf. [168]. In fact, we have then jLt1 .s/ Lt2 .s/j D ju.t1 C s/ u.t2 C s/j Z t1 Cs 0 D u ./ d t2 Cs

1

jt1 t2 j1 q ku0 kLq .t2 Cs;t1 CsIH/ : and integration with respect to s in the interval Œ0; l gives (10.57) with c depending on and l. We shall prove that there exists M > 0 such that k0 kL1 .0;lIH/ M

for all

2 Bl1 :

The formal a priori estimates that follow can be performed on the smooth in the time variable Galerkin approximations vın .t/, n D 1; 2; 3; : : :, of the regularized problem (10.31)–(10.32), as in [148]. The obtained estimates are preserved by solutions vı .t/ of problem (10.31)–(10.32), with bounds independent of ı when ı ! 0, and in the end, by solutions v.t/ of Problem 10.1.

230


Let us consider the solution v D v.t/ of problem (10.31)–(10.32) (we drop the subscript ı for short),

dv.t/ ; C a.v.t/; / C b.v.t/; v.t/; / C h˚ı0 .v.t//; i dt D a.; / b.; v.t/; / b.v.t/; ; /;

(10.58)

with the initial condition v.0/ D v0 : Our aim is to derive, following the method used in [148], two a priori estimates following from (10.58). To get the first one, set D vt (vt D v 0 ) in (10.58). We obtain jvt j2 C

d d kvk2 C ˚ı .v/ D hL .v/; vt i b.v; v; vt /: dt 2 dt

(10.59)

Using the Ladyzhenskaya inequality (10.30) we have hL .v/; vt i c1 .kvt k C kvkjvt j1=2 kvt k1=2 C jvj1=2 kvk1=2 jvt j1=2 kvt k1=2 /: (10.60) Now, we differentiate (10.58) with respect to the time variable and set D vt , to get 1d jvt j2 C kvt k2 b.vt ; ; vt / b.vt ; v; vt /; 2 dt

(10.61)

as b.; vt ; vt / D 0, b.v; vt ; vt / D 0, and, cf. [91, 188],

d 0 ˚ .v/; vt 0: dt ı

Using Lemma 10.1 and the Ladyzhenskaya inequality (10.30) to estimate the righthand side of (10.61) we obtain d jvt j2 C kvt k2 c2 kvk2 jvt j2 : dt

(10.62)

Now, we multiply (10.62) by t2 to get d 2 2 .t jvt j / C t2 kvt k2 c2 kvk2 .t2 jvt j2 / C 2tjvt j2 : dt

(10.63)

10.3 Planar Shear Flows with Generalized Tresca Type Friction Law

231

To get rid of the last term on the right-hand side we add to (10.63) Eq. (10.59) multiplied by 2t. After simple calculations and using (10.60) we obtain d 2 2 .t jvt j C tkvk2 C 2t˚ı .v// C t2 kvt k2 dt 2tc1 .kvt k C kvkjvt j1=2 kvt k1=2 C jvj1=2 kvk1=2 jvt j1=2 kvt k1=2 / C c2 kvk2 .t2 jvt j2 / C kvk2 C 2˚ı .v/ C c3 jvj1=2 kvk3=2 .tjvt j/1=2 .tkvt k/1=2 :

(10.64)

Define y D t2 jvt j2 Ctkvk2 C2t˚ı .v/. Using the Young inequality to the right-hand side of (10.64) and observing that ˚ı .v/ C.kvk2 C 1/ for 0 < ı 1, we obtain the inequality of the form t2 d y.t/ C kvt k2 C1 .t/y.t/ C C2 .t/; dt 2 where the coefficients Ci ./, i D 1; 2, are locally integrable and do not depend on ı. They are also independent of the initial conditions for v in a given bounded sets in H. This proves that the time derivative of solutions of the regularized problems is uniformly bounded with respect to ı in L1 .; TI H/ \ L2 .; TI V/ for all intervals Œ; T, 0 < < T. As a consequence, this property holds for all 2 Bl1 . In view of the above considerations this ends the proof of the existence of an exponential attractor. t u

10.3 Planar Shear Flows with Generalized Tresca Type Friction Law The problem considered in this section is based on the results of Sect. 10.2, cf. also [162]. As we show, the arguments used there can be generalized to a class of problems with the friction coefficient dependent on the slip rate.

10.3.1 Classical Formulation of the Problem We will consider the planar flow of an incompressible viscous fluid governed by the equation of momentum vt v C .v r/v C rp D f

in ˝ RC ;

(10.65)

232


and the incompressibility condition div v D 0

in ˝ RC

(10.66)

in domain ˝ given by ˝ D fx D .x1 ; x2 / W 0 < x1 < L; 0 < x2 < h.x1 /g; with boundary @˝ D D [C [L , where D is the top part of the boundary given by D D f.x1 ; h.x1 // W x1 2 .0; L/g, C is the bottom part of the boundary given by C D .0; L/ f0g, and L is the lateral one given by L D f0; Lg .0; h.0//. The function h is smooth and L-periodic such that h.x1 / " > 0 for all x1 2 R with a constant " > 0. We will use the notation e1 D .1; 0/ and e2 D .0; 1/ for the canonical basis of R2 . Note that on C the outer normal unit vector is given by n D e2 . The unknowns are the velocity field v W ˝ RC ! R2 and the pressure field p W ˝ RC ! R, > 0 is the viscosity coefficient and f W ˝ ! R2 is the mass force density. The stress tensor T is related to the velocity and pressure through the linear constitutive law Tij D pıij C .vi;j C vj;i /. We are looking for the solutions of (10.65)–(10.66) which are L-periodic in the 2 ;t/ 2 ;t/ sense that v.0; x2 ; t/ D v.L; x2 ; t/, @v2 .0;x D @v2 .L;x , and p.0; x2 ; t/ D p.L; x2 ; t/ @x1 @x1 C for x2 2 Œ0; h.0/ and t 2 R . The first condition represents the L-periodicity of velocities, while the latter two ones, the L-periodicity of the Cauchy stress vector Tn on L in the space of divergence free functions. Moreover we assume that vD0

on

D RC :

(10.67)

On the contact boundary C we decompose the velocity into the normal component vn D v n and the tangential one v D v e1 . Note that since the domain ˝ is two-dimensional it is possible to consider the tangential components as scalars, for the sake of the ease of notation. Likewise, we decompose the stress on the boundary C into its normal component Tn D Tn n and the tangential one T D Tn e1 . We assume that there is no flux across C and hence vn D 0 on

C RC :

(10.68)

The boundary C is assumed to be moving with the constant velocity U0 e1 D .U0 ; 0/ which, together with the mass force, produces the driving force of the flow. The friction coefficient k is assumed to be related to the slip rate through the relation k D k.jv U0 j/, where k W RC ! RC . If there is no slip between the fluid and the boundary then the friction force is bounded by the friction threshold k.0/ v D U0 ) jT j k.0/

on

C RC ;

(10.69)


233

while if there is a slip, then the friction force is given by the expression v ¤ U0 ) T D k.jv U0 j/

v U0 jv U0 j

C RC :

on

(10.70)

Note that (10.69)–(10.70) generalize the Tresca law considered in Sect. 10.2, where k was assumed to be a constant. Here k depends on the slip rate, this dependence represents the fact that the kinetic friction is less than the static one, which holds if k is a decreasing function. Similar friction law is used, for example, in the study of the motion of tectonic plates, see [119, 190, 207] and the references therein. We make the following assumptions on the friction coefficient k: .k1/ k 2 C.RC I RC /, .k2/ k.s/ ˛.1 C s/ for all s 2 RC with ˛ > 0, .k3/ k.s/ k.r/ .s r/ for all s > r 0 with > 0. Note that the assumption that k has values in RC has a clear physical interpretation, namely it means that the friction force is dissipative. Finally, the initial condition for the velocity field is v.x; 0/ D v0 .x/

x 2 ˝:

for

(10.71)

In the next section we present a weak formulation of the problem in the form of an evolutionary differential inclusion with a suitable potential corresponding to the generalized Tresca condition.

10.3.2 Weak Formulation of the Problem To be able to define a weak solution of problem (10.65)–(10.71) we introduce some notation. Let VQ D fv 2 C1 .˝/2 W div v D 0 in ˝; v D 0 on D ;

v is L-periodic in x1 ;

vn D 0 on C g;



We define scalar products in H and V, respectively, by Z

Z .u ; v/ D

u.x/v.x/ dx ˝

and

.ru ; rv/ D

ru.x/W rv.x/ dx; ˝

234



jvj D .v; v/ 2

and

1

kvk D .rv ; rv/ 2 :

˜ We denote the trace operator from V to L2 .C /2 by and its norm by k k D k kL .VIL2 .C /2 / . Moreover, let, for u; v, and w in V, a.u ; v/ D .ru ; rv/

and

b.u ; v ; w/ D ..u r/v ; w/:

Finally, we define the functional j W R ! R corresponding to the generalized Tresca condition (10.69)–(10.70) by Z

jvU0 j

j.v/ D

k.s/ ds:

(10.72)

0

Note that the functional j is not necessarily convex, but it is locally Lipschitz. By @j we will denote its Clarke subdifferential (see Sect. 3.7). The variational formulation of problem (10.65)–(10.71) can be derived by the calculation analogous to the proof of Proposition 10.1. Problem 10.2. Given v0 2 H and f 2 H, find v W RC ! H such that: 2 2 .i/ v 2 C.RC I H/ \ Lloc .RC I V/, with vt 2 Lloc .RC I V 0 /, C .ii/ for all in V and for almost all t 2 R , the following equality holds

hvt .t/; i C a.v.t/; / C b.v.t/; v.t/; / C ..t/; /L2 .C / D .f ; /; (10.73) 2 2 with .t/ 2 [email protected] for a.e. t 2 RC , where for w 2 L2 .C / we denote by [email protected]/ .t// the set of all functions 2 L2 .C / such that .x/ 2 @j.w.x// for a.e. x 2 C , .iii/ the following initial condition holds:

v.0/ D v0 :

(10.74)

The above definition is justified by the assertion .i/ of the following lemma. Lemma 10.3. Under assumptions .k1/–.k3/ the functional j defined by (10.72) satisfies the following properties: .i/ j is locally Lipschitz and conditions (10.69)–(10.70) are equivalent to T 2 @j.v /; .ii/ jj ˛.1 Q C jwj/ for all w 2 R and 2 @j.w/, ˛Q > 0,

(10.75)


235

.iii/ m.@j/ , where m.@j/ is the one sided Lipschitz constant defined by m.@j/ D

inf u;v2R;u¤v; [email protected]/;[email protected]/

. / .u v/ ; ju vj2

.iv/ for all w 2 R and 2 @j.w/, w ˇ.1 C jwj/, with a constant ˇ > 0. Proof. .i/ The fact that j is locally Lipschitz follows from the continuity of k and the direct calculation. To see that (10.69)–(10.70) is equivalent to (10.75) observe that for v ¤ U0 the functional j is continuously differentiable on a neighborhood vU0 of v and its derivative is given as j0 .v/ D k.jv U0 j/ jvU , whence (10.70) is 0j equivalent to (10.75) by the characterization of the Clarke subdifferential given in Theorem 3.19. The same characterization implies that (10.69) is equivalent to (10.75) if v D U0 . Assertion .ii/ follows in a straightforward way from .k2/. Assertion .iii/ can be obtained by a following computation. Let 2 @j.u/, 2 @j.v/, where u ¤ U0 and v ¤ U0 . Then u U0 v U0 k.jv U0 j/ .u v/ . / .u v/ D k.ju U0 j/ ju U0 j jv U0 j D k.ju U0 j/ju U0 j k.ju U0 j/ k.jv U0 j/

.u U0 / .v U0 / ju U0 j

.u U0 / .v U0 / C k.jv U0 j/jv U0 j jv U0 j

.k.ju U0 j/ k.jv U0 j// .ju U0 j jv U0 j/ .ju U0 j jv U0 j/2 ju vj2 : The calculation in the case, when either u D U0 or v D U0 is similar. Proof of assertion .iv/ is also straightforward. Indeed, for w D 0 the assertion obviously holds. Let w ¤ 0 and 2 @j.w/. We have, by .k1/–.k2/, w D k.jw U0 j/

w U0 .w U0 C U0 / jw U0 j

k.jw U0 j/jw U0 j k.jw U0 j/jU0 j ˛.1 C jwj C jU0 j/jU0 j; and .iv/ follows. The proof is complete.

t u

Remark 10.1. Observe that assertion .iii/ of Lemma 10.3 is equivalent to the claim 2 that the functional j is semiconvex, i.e., the functional s ! j.s/ C s2 is in fact convex (see Definition 10 in [18]).

236


10.3.3 Existence and Properties of Solutions We begin with some estimates that are satisfied by the solutions of Problem 10.2. We define the auxiliary operators A W V ! V 0 and B W V ! V 0 by hAu; vi D a.u; v/ and hBu; vi D b.u; u; v/. Lemma 10.4. Let v be a solution of Problem 10.2. Then, for all t 0, max jv.s/j2 C

s2Œ0;t

Z

t 0

Z

t 0

kv.s/k2 ds C.t; jv0 j/;

kv 0 .s/k2V 0 ds C.t; jv0 j/;

(10.76) (10.77)

where C.t; jv0 j/ is a nondecreasing function of t and jv0 j. Proof. Take D v.t/ in (10.73). Since, for v 2 V it is b.v; v; v/ D 0, we get, with an arbitrary " > 0, 1d jv.t/j2 C kv.t/k2 jf jjv.t/j C k.t/kL2 .C / kv .t/kL2 .C / 2 dt "jv.t/j2 C

3 jf j2 2 C ˛kv.t/k Q Q 1 .C /; L2 .C / C ˛m 4" 2

where we used (ii) of Lemma 10.3 and m1 .C / D L is the one-dimensional boundary measure of C . Denoting Z D V \ H 1ı .˝/2 with some small ı 2 0; 12 endowed with the norm H 1ı .˝/2 , observe that the embedding V Z is compact and the trace map C W Z ! L2 .C /2 is continuous. We denote kC k D kC kL .ZIL2 .C /2 / . From the Ehrling lemma (cf. Lemma 3.17) we get, for an arbitrary " > 0 independent on v 2 V and C."/ > 0 that kvk2Z "kvk2 C C."/jvj2 : Hence, noting that the Poincaré inequality 1 jvj2 kvk2 is valid for v 2 V, we get d 2" jf j jv.t/j2 C 2kv.t/k2 kv.t/k2 C dt 1 2" Q C k2 C."/jv.t/j2 C 2˛L: Q C 3˛k Q C k2 "kv.t/k2 C 3˛k


237

It is clear that we can choose " small enough such that d jv.t/k2 C kv.t/k2 C1 C C2 jv.t/j2 ; dt with C1 ; C2 > 0. Using the Gronwall inequality (cf. Lemma 3.20) we get (10.76). For the proof of (10.77) observe that for u 2 V, kBukV 0 Cjujkuk. Indeed, by the Hölder inequality and the Ladyzhenskaya inequality, jhBu; zij D jb.u; u; z/j D jb.u; z; u/j kuk2L4 .˝/2 jrzj Ckukjujkzk

for

u; z 2 V:

(10.78)

Hence (10.77) follows by (10.76) and the straightforward computation that uses the assertion .ii/ of Lemma 10.3 to estimate the multivalued term. t u We formulate and prove the theorem on existence of solutions to Problem 10.2. Theorem 10.8. For any u0 2 H Problem 10.2 has at least one solution. Proof. Since the proof of solution existence, based on the Galerkin method, is standard and quite long, we provide only its main steps. R1 Let % 2 C01 ..1; 1// be a mollifier such that 1 %.s/ ds D 1 and %.s/ 0. %k W R ! R by %k .x/ D k%.kx/ for k 2 N and x 2 R. Then supp %k We1define k ; 1k . We consider jk W R ! R defined by the convolution Z %k .s/j.r s/ ds for r 2 R: (10.79) jk .r/ D R

Note that jk 2 C1 .R/. From the assertion .B/ of Theorem 3.21 it follows that for all n 2 N and r 2 R we have jj0k .r/j ˛.1 C jrj/, where ˛ is different from ˛Q in .ii/ of Lemma 10.3 above but independent of k. Let us furthermore take the sequence Vk of finite dimensional spaces such that Vk D spanfz1 ; : : : ; zk g, where fzi g are orthonormal in H eigenfunctions of the Stokes operator with the Dirichlet and periodic boundary conditions given in the definition S1 of the space V. Then fVk g1 kD1 Vk D V. kD1 approximate V from inside, i.e., 0 We denote by P W V ! V the orthogonal projection on V defined as Pk u D k k k Pk hu; z iz . Using Theorem 3.17 we conclude that P v ! v strongly in V 0 for i i k iD1 0 any v 2 V . We formulate the regularized Galerkin problems for k 2 N. Find vk 2 C.RC I Vk / such that for a.e. t 2 RC , vk is differentiable and hvk0 .t/; i C a.vk .t/; / C b.vk .t/; vk .t/; / C .j0k .vn .t/.//; /L2 .C / D .f ; /; vk .0/ D Pk v0 for a.e. t 2 RC and all 2 Vk .

(10.80) (10.81)

238


Note that due to the fact that j0k ; a; b are smooth, the solution of (10.80)–(10.81), if it exists, is a continuously differentiable function of time variable, with values in Vk . We first show that if vk solves (10.80) then estimates analogous to the ones in Lemma 10.4 hold. Taking vk .t/ in place of in (10.80) and using the argument similar to that in the proof of (10.76) in Lemma 10.4 we obtain that, for a given T > 0, kvk kL1 .0;TIH/ const;

(10.82)

kvk kL2 .0;TIV/ const:

(10.83)

Now, denoting by W V ! L2 .C / the map v D . v/ and by the adjoint to , we observe that (10.80) is equivalent to the following equation in V 0 , vk0 .t/ C Avk .t/ C Pk B.vk .t// C Pk j0k .vk .t/.// D Pk f :

(10.84)

Since, by Theorem 3.17, for w 2 V 0 we have kPk wkV 0 kwkV 0 , then, using the estimate kBwkV 0 Cjwjkwk valid for w 2 V, the growth condition on j0k as well as (10.82) and (10.83), from Eq. (10.84) we get kvk0 kL2 .0;TIV 0 / const:

(10.85)

The existence of the solution to the Galerkin problem (10.80) follows by the Carathéodory theorem and estimates (10.82)–(10.83). Since the bounds (10.82), (10.83), and (10.85) are independent of k then, for a subsequence, we get vk ! v

weakly in L1 .0; TI H/;

vk ! v

weakly in L2 .0; TI V/;

vk0 ! v 0

weakly in L2 .0; TI V 0 /;

vk ! v vk ! v

strongly in L2 .0; TI L2 .C //;

(10.86)

strongly in L2 .0; TI H/;

(10.87)

where (10.86) is a consequence of the Aubin–Lions compactness theorem used for the spaces V Z V 0 and (10.87) is a consequence of the Aubin–Lions compactness theorem used for the spaces V H V 0 . From (10.83) and the growth condition on j0k it follows that kj0k .vn /kL2 .0;TIL2 .C // const: Hence, perhaps for another subsequence, j0k .vk / !

weakly in

L2 .0; TI L2 .C //:


239

In a standard way (see, for example, Sects. 7.4.3 and 9.4 in [197]) we can pass to the limit in (10.84) and obtain that v and satisfy (10.73) for a.e. t 2 .0; T/. The 2 assertion .C/ of Theorem 3.21 implies that .t/ 2 [email protected] for a.e. t 2 .0; T/. .t// We shall show that v.0/ D v0 in H. We have, for t 2 .0; T/, Z vk .t/ D Pk v0 C

0

t

Z

vk0 .s/ ds;

v.t/ D v.0/ C

t 0

v 0 .s/ ds;

where the equalities hold in V 0 . Take 2 V. Then Z

T 0

hvk .t/ v.t/; i dt Z

D 0

T

Z hPk v0 v.0/; i dt C Z

D ThPk v0 v.0/; i C

0

T

T 0

Z 0

t

.vk0 .s/ v 0 .s// ds; dt

hvk0 .s/ v 0 .s/; .T s/i ds:

Since vk ! v weakly in L2 .0; TI V/ and vk0 ! v 0 weakly in L2 .0; TI V 0 /, it follows that Pk v0 ! v.0/ weakly in V 0 . We know that Pk v0 ! v0 strongly in V 0 and hence v.0/ D v0 . Finally, the solution can be extended from Œ0; T to the whole RC . Indeed, we can take the value of the solution at the time point T as an initial condition and solve the resulting problem, obtaining another solution on Œ0; T. Then, we concatenate the two solutions, obtaining the solution defined on the interval Œ0; 2T. Repeating this procedure we get the solution on the whole RC (see, for example, [93], Theorem 2 in Sect. 9.2.1). t u Remark 10.2. In the proof of Theorem 10.8 we used only assertions .i/ and .ii/ of Lemma 10.3 and therefore the theorem holds for a class of potentials j that satisfy these two conditions and are not necessarily given by (10.72). This is a new result of independent interest since it weakens the conditions required for the existence of solutions provided, for example, in [174]. Indeed, the sign condition (see H(j)(iv) p. 583 in [174]) is not needed here for the existence proof. Next lemma shows that assertion .iii/ of Lemma 10.3 gives the Lipschitz continuity of the solution map on bounded sets as well as the solution uniqueness. Lemma 10.5. The solution of Problem 10.2 is unique and if v, w are two solutions of Problem 10.2 with the initial conditions v0 ; w0 , respectively, then for any t > s 0 we have jv.t/ w.t/j D.t; jw0 j/jv.s/ w.s/j; where D.t; jw0 j/ > 0 is a nondecreasing function of t; jw0 j.

(10.88)

240


Proof. Let v and w be two solutions of Problem 10.2 with the initial conditions v0 and w0 . We denote u.t/ D v.t/ w.t/. Subtracting (10.73) for v and w, respectively, and taking u.t/ as a test function we get, for a.e. t 2 RC , 1d ju.t/j2 C ku.t/k2 C b.u.t/; w.t/; u.t// C ..t/ .t/; v.t/ w.t/ /L2 .C / D 0; 2 dt 2 2 and .t/ 2 [email protected] . where .t/ 2 [email protected] .t// .t// Using estimate (10.78) to estimate the convective term and also assertion .iii/ in Lemma 10.3 to estimate the multivalued one we get, for a.e. t 2 RC ,

1d ju.t/j2 C ku.t/k2 ku .t/k2L2 .C / Cju.t/jku.t/kkw.t/k: 2 dt

(10.89)

As in the proof of Lemma 10.4 we use the fact that the embedding V Z is compact and the trace C W Z ! L2 .C /2 is continuous. We get, for a.e. t 2 RC , 1d ju.t/j2 C ku.t/k2 "ku.t/k2 C C."/ju.t/j2 kw.t/k2 2 dt C kC k2 ."ku.t/k2 C C."/ju.t/j2 /

(10.90)

with an arbitrary " > 0 and the constant C."/ > 0 independent on u, w, and t. By choosing " > 0 small enough we get d ju.t/j2 ju.t/j2 .C1 kw.t/k2 C C2 / dt

for a.e.

t 2 RC ;

with the constants C1 ; C2 > 0. The Gronwall inequality (cf. Lemma 3.20) gives ju.t/j2 ju.s/j2 eC2 .ts/CC1

Rt s

kw. /k2 d

ju.s/j2 eC2 tCC1

Rt 0

kw. /k2 d

:

Using (10.76) we get 1

1

jv.t/ w.t/j jv.s/ w.s/je 2 C2 tC 2 C1 C.t;jw0 j/ ;

(10.91)

and the proof of (10.88) is complete. Taking s D 0 and v.0/ D w.0/ in (10.88) we obtain the uniqueness. t u Now we shall prove that assertion .iv/ of Lemma 10.3 gives additional, dissipative, a priori estimates. Lemma 10.6. If v is a solution of Problem 10.2 with the initial condition v0 , then, jv.t/j jv0 jeC1 t C C2

for all

with positive constants C1 ; C2 independent of t; v0 .

t 0;


241

Proof. We proceed as in the proof of (10.76) in Lemma 10.3. Taking the test function D v.t/ in (10.73) we get 1d jv.t/j2 C kv.t/k2 C ..t/; v .t//L2 .C / jf jjv.t/j 2 dt for a.e. t 2 RC . We use assertion .iv/ of Lemma 10.3 and the Poincaré inequality 1 jvj2 kvk2 to get 1d 1 jv.t/j2 C kv.t/k2 ˇL 1 C ˇ"1 kv .t/k2L2 .C / 2 dt 4"1

jf j2 "2 C kv.t/k2 4"2 1

for a.e. t 2 RC , with an arbitrary "1 > 0 and "2 > 0. Using the trace inequality and the Poincaré inequality again, we can choose "1 and "2 small enough to get d jv.t/j2 C 1 jv.t/j2 C dt for a.e. t 2 RC , where C > 0 is a constant. Finally, the assertion follows from the Gronwall inequality (cf. Lemma 3.20). t u

10.3.4 Existence of Finite Dimensional Global Attractor In this subsection we prove the following theorem. Theorem 10.9. The semigroup fS.t/gt0 associated with Problem 10.2 has a global attractor A H of finite fractal dimension in the space H. Proof. In view of Theorem 10.2 we need to verify assumptions .A1/–.A8/. We set Y D V, X D H, Z D V 0 , YT D fu 2 L2 .0; TI V/ W u0 2 L2 .0; TI V 0 /g for T > 0, and l > 0, Xl D L2 .0; lI H/. Assumption .A1/. From Theorem 10.8 it follows that for any u0 2 H there exists at least one solution of Problem 10.2. This solution restricted to the interval .0; T/ belongs to YT and C.Œ0; TI H/. Uniform bounds in YT follow directly from Lemma 10.4. Assumption .A2/. In view of Lemma 10.6 the closed ball BH .0; C2 C 1/ absorbs S H in finite time all bounded sets in H. Let B0 D t0 S.t/BH .0; C2 C 1/ . Obviously this set is closed in H (we will need this fact to verify the assumption .A5/ below). From Lemma 10.6 it follows that B0 is bounded. Let us assume that u 2 B0 and v D S.t/u. Then, for a sequence uk 2 S.tk /BH .0; C2 C 1/ with certain tk we have

242


uk ! u in H. From Lemma 10.5 it follows that S.t/uk ! S.t/u D v in H. But S.t/uk 2 S.t C tk /BH .0; C2 C 1/ B0 . Hence, from the closedness of B0 it follows that v 2 B0 and the assertion is proved. In the sequel of the proof the generic constant depending on B0 ; l and the problem data will be denoted by Cl;B0 . Assumption .A3/. Recall that the l-trajectory is a restriction of any solution of Problem 10.2 to the time interval Œ0; l. The fact that assumption .A3/ holds follows immediately from Lemma 10.5. Assumption .A4/. Define the shift operator Lt W Xl ! Xl by .Lt /.s/ D u.s C t/ for s 2 Œ0; l, where u is the unique solution on Œ0; l C t such that ujŒ0;l D . We will prove that this operator is continuous on Bl0 for all t 0 and l > 0, where Bl0 is the set of all l-trajectories with the initial condition in the set B0 from assumption .A2/. Let u; v be two solutions with the initial conditions u0 ; v0 2 B0 . By Lemma 10.5 we have Z l Z l jv.s C t/ u.s C t/j2 ds D2 .t C l; jB0 j/ jv.s/ u.s/j2 ds; 0

0

where jB0 j D supv2B0 jvj. Hence, denoting 1 D ujŒ0;l and 2 D vjŒ0;l , we get kLt 1 Lt 2 k2Xl D2 .t C l; jB0 j/k1 2 k2Xl ;

(10.92)

and the proof of .A4/ is complete. Xl

Assumption .A5/. We need to prove that, for some > 0, L .Bl0 / Bl0 . Let u0 2 B0 . From the fact that for all > 0, S. /u0 2 B0 , it follows that L .Bl0 / Bl0 . To obtain the assertion we must show that Bl0 is closed in Xl . To this end assume that uk is a sequence of solutions to Problem 10.2 such that uk .0/ 2 B0 and that k ! strongly in L2 .0; lI H/;

(10.93)

where k D uk jŒ0;l . From .A1/ it follows that k !

weakly in L2 .0; lI V/;

(10.94)

0k ! 0

weakly in L2 .0; lI V 0 /:

(10.95)

We shall show that 2 Bl0 . Let us first prove that .0/ 2 B0 . Indeed, from (10.93) it follows that, for a subsequence, k .t/ ! .t/ strongly in H for a.e. t 2 .0; l/. Since k .t/ 2 B0 for all k 2 N and all t 2 Œ0; l, then .t/ belongs to the closed set B0 for a.e. t 2 .0; l/. The closedness of B0 and the fact that 2 C.Œ0; lI H/ imply that .t/ 2 B0 for all t 2 Œ0; l and, in particular, .0/ 2 B0 . We need to prove that satisfies (10.73) for a.e. t 2 .0; l/. It is enough to show that for all w 2 L2 .0; lI V 0 /,


Z

l 0

h0 .t/; w.t/i dt C Z

l

C

Z

l

243

hA.t/; w.t/i dt

0

Z

l

hB.t/; w.t/i dt C

0

0

Z ..t/; w .t//L2 .C / dt D

l 0

.f ; w.t// dt

(10.96)

2 with .t/ 2 S@j. for a.e. t 2 .0; l/. From the fact that k satisfies (10.73) for a.e. .t// t 2 .0; l/ we have

Z 0

l

h0k .t/; w.t/i dt C Z C 0

l

Z

l 0

hAk .t/; w.t/i dt Z

hBk .t/; w.t/i dt C

0

l

Z .k .t/; w .t//L2 .C / dt D

0

l

.f ; w.t// dt; (10.97)

2 with k .t/ 2 S@j. for a.e. t 2 .0; l/. Then, from the growth condition .ii/ in k .t// p p Lemma 10.3 it follows that kk kL2 .0;lIL2 .C // 2lL˛Q C 2˛k Q k kL2 .0;lIL2 .C // . Since k is bounded in L2 .0; lI V/, then k is bounded in L2 .0; lI L2 .C // and hence k is bounded in the same space. Therefore, for a subsequence,

k !

weakly in L2 .0; lI L2 .C //:

(10.98)

Now (10.93), (10.94), and (10.98) are sufficient to pass to the limit in all the terms 2 for a.e. in (10.97) and thus to prove (10.96). It remains to show that .t/ 2 S@j. n .t// 1ı 2 1ı t 2 .0; l/. Let Z D V \H .˝/ be equipped with H topology, where ı 2 0; 12 . Then for the triple V Z V 0 we can use the Aubin–Lions compactness theorem and conclude from (10.93) and (10.94) that k ! strongly in L2 .0; lI Z/. Now, since the trace operator C W Z ! L2 .C /2 is linear and continuous, and so is the corresponding Nemytskii operator, k !

strongly in

L2 .0; lI L2 .C //:

(10.99)

We use assertion .H2/ in Theorem 3.24 to deduce that .x; t/ 2 @j. .x; t// a.e. in C .0; l/. This completes the proof of .A5/. Assumption .A6/. We define W D H01 .˝/2 \ V, where we equip W with the norm of V. Then V 0 W 0 . By the Aubin–Lions compactness theorem the space Wl D f 2 L2 .0; lI V/ W 0 2 L1 .0; TI W 0 /g is embedded compactly in Xl . We must show that the shift operator L W Xl ! Wl Xl

is Lipschitz continuous on Bl1 D L .Bl0 / for some > 0. In fact, we shall prove that L is Lipschitz continuous on the larger set Bl0 for D l. Indeed, let w; v be two solutions of Problem 10.2 starting from B0 . Denote u D v w. Then, by (10.90),

244


d ju.t/j2 C ku.t/k2 Cju.t/j2 .1 C kw.t/k2 / dt

for a.e.

t 2 RC ;

where the constant C depends only on the problem data. We fix s 2 .0; l/ and integrate the last inequality over interval .s; 2l/ to obtain ju.2l/j2 C

Z

2l

ku.t/k2 dt C

s

Z

2l

ju.t/j2 .1 C kw.t/k2 / dt C ju.s/j2 :

s

Using (10.91) we get Z

2l

Z ku.t/k ju.s/j 2Cl;B0 l C Cl;B0 2

2

0

l

2l

kw.t/k dt C 1 : 2

Since, by .A1/, Z

2l

0

kw.t/k2 dt Cl;B0 ;

it follows that Z

2l

ku.t/k2 dt Cl;B0 ju.s/j2 :

l

Integrating this inequality over interval .0; l/ with respect to the variable s it follows that Z l

2l

Z

2

ku.t/k dt Cl;B0

l

l

ju.s/j2 ds;

0

and therefore r kLl v Ll wkL2 .0;lIV/

Cl;B0 kv wkXl : l

It remains to prove that k.Ll v Ll w/0 kL2 .0;lIW 0 / Cl;B0 kv wkXl : In view of (10.100) it is sufficient to prove that k.v w/0 kL2 .0;lIW 0 / Cl;B0 kv wkL2 .0;lIV/ :

(10.100)


245

To this end, we subtract (10.73) for w and v, respectively, and take 2 W as a test function. We get hu0 .t/; i C hAu.t/; i C hBv.t/ Bw.t/; i D 0: We have, for a constant k > 0 dependent only on the problem data, hBv.t/ Bw.t/; i D b.u.t/; w.t/; / C b.w.t/; u.t/; / C b.u.t/; u.t/; / k.ku.t/kkw.t/k C ku.t/k2 /kk k.2kw.t/k C kv.t/k/ku.t/kkk; whence, for a constant C > 0 dependent only on the problem data, ku0 .t/kW 0 D

sup 2W;kkD1

jhAu.t/; i C hBv.t/ Bw.t/; ij

kAu.t/kV 0 C kBv.t/ Bw.t/kV 0 C.1 C kw.t/k C kv.t/k/ku.t/k: It follows that ku0 kL1 .0;lIW 0 / CkukL2 .0;lIV/

s Z

l 0

.1 C kw.t/k C kv.t/k/2 dt:

The assertion follows from .A1/. Assumptions .A7/ and .A8/. Since .A8/ implies .A7/ we prove only .A8/. We shall show that the mapping e W Xl ! X defined as e./ D .l/ is Lipschitz on Bl0 . Indeed, by (10.91), for any two solutions v; w of Problem 10.2 such that their initial conditions belong to B0 , we get jw.l/ v.l/j Cl;B0 jw.s/ v.s/j for all

s 2 .0; l/:

Integrating the square of this inequality over s 2 .0; l/ we obtain 2 2 ljw.l/ v.l/j2 Cl;B 0 kw vkXl ;

whence Cl;B0 jw.l/ v.l/j p kw vkXl ; l and .A8/ holds.

246


This completes the proof Theorem 10.9 about the existence of the global attractor of a finite fractal dimension. t u In the following subsection we shall prove the existence of an exponential attractor.

10.3.5 Existence of an Exponential Attractor Our goal is to prove the following: Theorem 10.10. The semigroup fS.t/gt0 associated with Problem 10.2 has an exponential attractor in the space H. Proof. Since we have shown, in the proof of Theorem 10.9, that assumptions .A1/– .A8/ hold, in view of Theorem 10.3 it is enough to prove .A9/ and .A10/. Assumption .A9/. Lipschitz continuity of Lt W Xl ! Xl , uniform with respect to t 2 Œ0; for all > 0, follows immediately from inequality (10.92). Assumption .A10/. It remains to prove that the inequality kLt1 Lt2 kXl cjt1 t2 jı holds for all t0 > 0, 0 t2 t1 t0 and 2 Bl1 with ı 2 .0; 1 and c > 0. Xl

Since Bl1 D L .Bl0 / for certain > 0, it is enough to prove the assertion for 2 L .Bl0 /. Let v be the unique solution of Problem 10.2 on Œ0; C l C t0 with v.0/ 2 B0 such that vjŒ; Cl D . It is enough to obtain the uniform bound on v 0 in the space L1 .; C l C t0 I H/. Indeed, s kLt1 Lt2 kXl D

Z 0

l

jv. C s C t2 / v. C s C t1 /j2 ds

s Z l Z 0

CsCt2

CsCt1

2 jv 0 .r/j dr

ds

p jt1 t2 j lkv 0 kL1 .;ClCt0 IH/ : The a priori estimate will be computed for the smooth solutions of the regularized Galerkin problem (10.80)–(10.81) formulated in the proof of Theorem 10.8. We shall show that, for the Galerkin approximations vk , for any T > , the functions vk0 are bounded in L1 .; TI H/ independently of k and, in consequence, the desired estimate will be preserved by their limit v, a solution of Problem 10.2. The argument follows the lines of the proof of Theorem 10.7 (compare [148]). First we take D vk0 .t/ in (10.80) which gives us


jvk0 .t/j2 C

247

1d kvk .t/k2 C b.vk .t/; vk .t/; vk0 .t// 2 dt Z d C jk .vk .x; t// dS D .f ; vk0 .t//: dt C

Estimating the convective term by using the Ladyzhenskaya inequality (10.30) we get jvk0 .t/j2 C

d d kvk .t/k2 C 2 dt dt

Z C

1

jk .vk .x; t// dS 3

1

(10.101) 1

jf j2 C Cjvk .t/j 2 kvk .t/k 2 jvk0 .t/j 2 kvk0 .t/k 2 ; where C > 0 is a constant. Now, a straightforward and technical calculation that uses the Fatou lemma, the Lebourg mean value theorem (cf. Theorem 3.20) and the conditions .ii/ and .iii/ in Lemma 10.3, shows, that for all r1 ; r2 2 R, .j0k .r2 / j0k .r1 //.r2 r1 / jr2 r2 j2 ; where the constant is the same as in .iii/ of Lemma 10.3. Hence, since vk is a continuously differentiable function of time variable,

d 0 0 0 jk .vk .x; t// vk .x; t/ jvk .x; t/j2 dt

for all .x; t/ 2 ˝ .0; T/: (10.102)

We differentiate both sides of (10.80) with respect to time and take D vk0 .t/, which gives us 1d 0 2 jv .t/j C kvk0 .t/k2 C 2 dt k

Z C

d 0 0 jk .vk .x; t// vk .x; t/ dS dt

C b.vk0 .t/; vk .t/; vk0 .t// C b.vk .t/; vk0 .t/; vk0 .t// D 0: Using (10.102) and the fact that b.vk .t/; vk0 .t/; vk0 .t// D 0 we get 1d 0 2 0 jv .t/j C kvk0 .t/k2 kvk .t/k2L2 .C / C b.vk0 .t/; vk .t/; vk0 .t// 0: 2 dt k Proceeding exactly as in the proof of (10.89) we get, for an arbitrary " > 0 and a constant C."/ > 0, 1d 0 2 jv .t/j C kvk0 .t/k2 C."/kvk .t/k2 jvk0 .t/j2 2 dt k C "kvk0 .t/k2 C C."/jvk0 .t/j2 ;

248


whence d 0 2 jv .t/j C kvk0 .t/k2 Cjvk0 .t/j2 .kvk .t/k2 C 1/: dt k Multiplying this inequality by t2 we get d 2 0 2 .t jvk .t/j / C t2 kvk0 .t/k2 Ct2 jvk0 .t/j2 .kvk .t/k2 C 1/ C 2tjvk0 .t/j2 : dt We add this inequality to inequality R (10.101) multiplied by 2t to obtain, after simple calculations that use the fact that C jk .vk .x; t// dS C.1 C kvk .t/k2 /, Z d 2 0 2 t jvk .t/j C 2tkvk0 .t/k2 C 4t jk .vk .x; t// dS C t2 kvk0 .t/k2 dt C C1 kvk .t/k2 .tjvk0 .t/j/2 C C2 .tjvk0 .t/j/2 C C3 C C4 kvk .t/k2 1

3

1

1

C C5 jvk .t/j 2 kvk .t/k 2 .tjvk0 .t/j/ 2 .tkvk0 .t/k/ 2 ; with Ci > 0 for i D 1; : : : ; 5 independent on k and initial data. Denoting yk .t/ D t2 jvk0 .t/j2 C 2tkvk0 .t/k2 C 4t

Z C

jk .vk .x; t// dS;

using the bound of Lemma 10.6 for jvk .t/j, Young inequality, and the fact that jk assumes nonnegative values, we get d yk .t/ C t2 kvk0 .t/k2 C6 kvk .t/k2 C C2 yk .t/ C C3 C C7 kvk .t/k2 ; dt 2 where C6 ; C7 > 0. Finally, the Gronwall inequality (cf. Lemma 3.20) and the bound RT in (10.76) of Lemma 10.4 for 0 kvk .s/k2 ds yield the uniform boundedness of vk0 in L1 .; TI H/ \ L2 .; TI V/ for all intervals Œ; T, 0 < < T. This completes the proof of .A10/ and thus the existence of an exponential attractor for the semigroup fS.t/gt0 associated with Problem 10.2. t u

10.4 Comments and Bibliographical Notes Large time behavior of solutions of problems in contact mechanics is an important though somewhat neglected part of the theory. In fact, remarking on future directions of research in the field of contact mechanics, in their book [215], the authors wrote: “The infinite-dimensional dynamical systems approach to contact problems is virtually nonexistent. (. . . ) This topic certainly deserves further consideration.”


249

From the mathematical point of view, a considerable difficulty in analysis of contact mechanics problems, and dynamical problems in particular, comes from the presence of involved boundary constraints which are often modeled by multivalued boundary conditions of a subdifferential type and lead to a formulation of the considered problem in terms of a variational inequality or differential inclusion with, frequently, nondifferentiable boundary functionals. To establish the existence of the global attractor of a finite fractal dimension we used the method of l-trajectories as presented in [168] (cf. also [167]). This method appears very useful when one deals with variational inequalities, cf. [208], as it overcomes obstacles coming from the usual methods presented in [62, 69, 111, 197, 220]. One does not need to check directly neither compactness of the dynamics which results from the second energy inequality nor asymptotic compactness, cf., e.g., [28, 220], which results from the energy equation. In the case of variational inequalities it is sometimes not possible to get the second energy inequality and the differentiability of the associated semigroup due to the presence of nondifferentiable boundary functionals. While there are other methods to establish the existence of the global attractor, where the problem of the lack of regularity appears, such as, e.g., the approach based on the notion of the Kuratowski measure of noncompactness of bounded sets, where we do not need even the strong continuity of the semigroup associated with a given dynamical problem, cf., e.g., [240], the problem of a finite dimensionality of the attractor is more involved. The method of l-trajectories allows to prove the existence of an even more desirable object, the exponential attractor, for many problems for which there exists a finite dimensional global attractor [179]. It contains the global attractor and thus its existence implies the finite dimensionality of the global attractor itself. Its crucial property is an exponential rate of attraction of solution trajectories [94, 179]. The proof of the existence of an exponential attractor requires the solution to be regular enough to ensure the Hölder continuity of the semigroup in the time variable [168]. We established this property by providing additional a priori estimates of solutions. In the end we would like to mention some related problems. First, there is a question of the existence of global and exponential attractors for other contact problems with multivalued boundary conditions in form of Clarke subdifferentials. As concerns exponential attractors, the main difficulty seems to be produced by assumption .A10/. Usually (10.2) follows from some better regularity of the time derivative of the solution, e.g., it easily follows if we know that u0 2 Lq .; TI X/ for T > > 0 and some q > 1. While some regularity results for contact problems are known [91, 215], such a property is not known to hold for most cases, and hence a question of the solution regularity for contact problems still demand further study. In particular we needed the convexity of underlying potential in the Tresca case, and .k3/, the one sided Lipschitz condition, in the generalized Tresca case. These properties guaranteed us the solution uniqueness. It is, to our knowledge, an open problem, to obtain results on the global attractor fractal dimension for the problems without the solution uniqueness, for example, if we do not assume .k3/.

250


For some visco-plastic flows (e.g., Bingham flows) governed by variational inequalities (however, only with homogeneous or periodic boundary conditions) the regularity problem in question was solved, e.g., in [148, 211] and used to study the time asymptotics of solutions. In this context, there are other important open problems, namely these of the time asymptotics of solutions of the Navier–Stokes equations, visco-plastic, and other fluid models governed by evolution variational or even hemivariational inequalities (cf., e.g., [176]) that take into account involved boundary conditions coming from a variety of applications in mechanics. Note that for the dissipative second order (in time) problems with multivalued boundary conditions the situation can be totally different from that for the first order problems. Namely, even for the case of a very simple convex potential and the gradient system, it is possible to show that the global attractor can have infinite fractal dimension (see [126]).

11

Non-autonomous Navier–Stokes Equations and Pullback Attractors

Running water has within itself an infinite number of movements which are greater or less than its principal course. – Leonardo da Vinci

In this chapter we study the time asymptotics of solutions to the two-dimensional Navier–Stokes equations. In the first two sections we prove two properties of the equations in a bounded domain, concerning the existence of determining modes and nodes. Then we study the equations in an unbounded domain, in the framework of the theory of infinite dimensional non-autonomous dynamical systems and pullback attractors.

11.1 Determining Modes Our aim in this section is to prove that if a number of Fourier modes of two different solutions of the Navier–Stokes equations have the same asymptotic behavior as t ! 1 then, under the condition that the corresponding external forces have the same asymptotic behavior as t ! 1, the remaining infinite number of modes also have the same asymptotic behavior. This means that both solutions have the same time asymptotics. We shall consider first two-dimensional flows with homogeneous Dirichlet boundary conditions, in an open, bounded, and connected domain ˝ of class C2 . To be more precise, let @u C u C .u r/u C rp D f ; @t

div u D 0;

@v C v C .v r/v C rq D g; @t

div v D 0;

and


251

252

11 Non-autonomous Navier–Stokes Equations and Pullback Attractors

and let u.x; t/ D

1 X

uO k .t/!k .x/;

Pm u.x; t/ D

kD1

m X

uO k .t/!k .x/;

kD1

and v.x; t/ D

1 X

vO k .t/!k .x/;

Pm v.x; t/ D

kD1

m X

vO k .t/!k .x/;

kD1

where the functions !k , k D 1; 2; 3; : : :, are the eigenfunctions of the Stokes operator A, A!k D k !k , cf. Chap. 4. We assume that the forcing terms f and g, f ; g 2 L1 .0; 1I H/, have the same asymptotic behavior as time goes to infinity, Z ˝

jf .x; t/ g.x; t/j2 dx ! 0 as

t ! 1:

(11.1)

Definition 11.1. Let (11.1) hold. Then the first m modes associated with Pm are called the determining modes if the condition Z ˝

jPm u.x; t/ Pm v.x; t/j2 dx ! 0 as

t!1

implies Z ˝

ju.x; t/ v.x; t/j2 dx ! 0 as

t ! 1:

(11.2)

Remark 11.1. Relation (11.2) implies further a similar convergence in the norm of V. Let Z lim sup t!1

˝

jf .x; t/j2 dx

1=2

D F:

We shall give the explicit estimate of the number m of determining modes in terms of the Grashof number G, which is a nondimensional parameter, defined here as GD

F 1 2

We shall prove the following

for the space dimension

n D 2:

11.1 Determining Modes

253

Theorem 11.1. Suppose that m 2 N is such that m cG2 where G is the Grashof number, G D 1F 2 , and c is a constant depending only on the shape of ˝. Then the first m modes are determining (in the sense defined above) for the two-dimensional Navier–Stokes system with non-slip boundary conditions. Proof. The proof is based on the generalized Gronwall lemma (Lemma 3.23), cf. [99]. We shall apply the lemma to .t/ D jQm w.t/j2 , where w.t/ D u.t/ v.t/ is the difference of the two solutions and Qm D I Pm , showing that if m is sufficiently large and jPm w.t/j ! 0 together with jf .t/ g.t/j ! 0 as t ! 1 then (3.36) holds. We have the following equation for the difference of solutions, dw C Aw C B.w; u/ C B.v; w/ D f .t/ g.t/: dt Taking the inner product with Qm w in H we obtain 1d jQm wj2 C kQm wk2 C b.w; u; Qm w/ C b.v; w; Qm w/ 2 dt D .f .t/ g.t/; Qm w/:

(11.3)

We estimate the nonlinear terms as follows. Observe that b.w; u; Qm w/ D b.Pm w; u; Qm w/ C b.Qm w; u; Qm w/:

(11.4)

We estimate the first term on the right-hand side using the Ladyzhenskaya inequality (3.8) to get jb.Pm w; u; Qm w/j D jb.Pm w; Qm w; u/j c1 jPm wj1=2 kPm wk1=2 juj1=2 kuk1=2 kQm wk: The second term on the right-hand side of (11.4) is estimated similarly to the first term, obtaining jb.Qm w; u; Qm w/j c1 jQm wjkQm wkkuk

c21 jQm wj2 kuk2 C kQm wk2 : 2 2

Observe that jb.v; w; Qm w/j D jb.v; Pm w; Qm w/j c1 jvj1=2 kvk1=2 jPm wj1=2 kPm wk1=2 kQm wk:

254


For the right-hand side of (11.3) we have j.f .t/ g.t/; Qm w/j jf .t/ g.t/jjQm wj: From the above estimates we conclude that c2 1d jQm wj2 C kQm wk2 1 kuk2 jQm wj2 2 dt 2 2 c1 jPm wj1=2 kPm wk1=2 juj1=2 kuk1=2 kQm wk Cc1 jvj

1=2

1=2

kvk

jPm wj

1=2

1=2

kPm wk

(11.5)

kQm wk C jf .t/ g.t/jjQm wj:

We use mC1 jQm wj2 kQm wk2

(11.6)

in the second term on the left-hand side of (11.5) to obtain d.t/ C ˛.t/.t/ ˇ.t/; dt with ˛.t/ D mC1

c21 ku.t/k2 ; 2

and ˇ D 2 fthe right-hand side of (11.5)g: We shall check the conditions of the generalized Gronwall lemma (i.e., Lemma 3.23). The existence of uniquely determined u and v follows from Theorem 7.2. Remark 7.2 implies that both u and v belong to L2 .; TI D.A// for all 0 < < T < 1. Now, as f ; g 2 L1 .RC I H/ we use an argument based on the enstrophy estimate and the uniform Gronwall inequality, similar as in the proof of Theorem 7.4, to deduce that the solutions u.t/ and v.t/ are uniformly bounded in t away from zero in both H and V norms (for the initial conditions in H), see Exercise 7.25. Therefore, if jPm w.t/j ! 0 and jf .t/ g.t/j ! 0 as t ! 1 then from inequality (11.5) it follows that ˇ.t/ ! 0 as t ! 1, which implies condition (3.35). Second, from the energy equation 1 ju.t/j2 C 2

Z

t

t0

ku.s/k2 ds D

1 ju.t0 /j2 C 2

Z

t t0

.f .s/; u.s//ds;

11.1 Determining Modes

255

the boundedness of ju.t/j, and the Poincaré inequality we have 1 T

Z

tCT

ku.s/k2 ds

t

2 kf k2L1 .t;tCTIH/ 2 1

for large

T:

(11.7)

By (11.7), condition (3.34) is satisfied. We have also lim inf t!1

Z

1 T

tCT

˛./d m

t

m

c21 1 lim sup t!1 T

Z

tCT

ku. /k2 d

t

2c21 F 2 : 3 1

Thus, for large m, mC1 >

2c21 F 2 ; 4 1

(11.8)

and (3.33) is satisfied. In view of Lemma 3.23, .t/ ! 0 as t ! 1 for sufficiently large m. For m ! 1 we have m c01 1 m where c01 is a nondimensional constant (cf. [99]), and we can see that (11.8) is satisfied for m cG2 , where c is a nondimensional number depending only on the shape of ˝. t u Exercise 11.1. Prove inequality (11.6). Exercise 11.2. Prove (11.7). Determining Modes in the Space-Periodic Case In this case we can prove in a similar way a stronger result due to better properties of the trilinear form b. Theorem 11.2. Suppose that m 2 N is such that m cG where G is the Grashof number, G D 1F 2 , and c is a constant depending only on the shape of ˝ (i.e., the ratio between the periods in the two space directions). Then the first m modes are determining in the sense that Z ˝

jru.x; t/ rv.x; t/j2 dx ! 0 as

t ! 1:

for the two-dimensional Navier–Stokes system with periodic boundary conditions and vanishing space average. Proof. The proof is more involved than that of Theorem 11.1 and can be found in [99]. The idea is again to use the generalized Gronwall lemma to the function .t/ D Qm w.t/. We start with equation dw C Aw C B.w; u/ C B.v; w/ D f .t/ g.t/; dt

256


take the inner product with AQm w in H (we know that u.t/ 2 D.A/ for t > 0) to obtain 1d kQm wk2 C kQm Awk2 C b.w; u; AQm w/ C b.v; w; AQm w/ 2 dt D .f .t/ g.t/; AQm w/; and then proceed as above using further properties (as these in Exercises 11.3 and 11.4) of solutions of two-dimensional Navier–Stokes equations with spaceperiodic boundary conditions. t u Exercise 11.3. Prove that for u; v 2 D.A/ b.v; u; Au/ C b.u; v; Au/ C b.u; u; Av/ D 0: Exercise 11.4. Prove that 1 T

Z

tCT

jAu.s/j2 ds

t

2 kf k2L1 .t;tCTIH/ 2

for large T:

(cf. Exercise 11.6).

11.2 Determining Nodes In many practical situations the experimental data is collected from measurements at a finite number of nodal points in the physical space. A set of points in the physical space (in the domain filled by the fluid) is called a set of determining nodes if, whenever the difference between the measurements at those points of the velocity field of any two flows goes to zero as time goes to infinity, then the difference between those velocity fields goes to zero uniformly on the domain (in a certain norm). We shall prove that there exists a finite number of determining nodes and shall estimate their lowest number. Let ˝ be an open, bounded, and connected two-dimensional domain of class C2 . We consider two velocity fields u D u.x; t/ and v D v.x; t/ satisfying in ˝ the Navier–Stokes equations, @u C u C .u r/u C rp D f ; @t

div u D 0;

@v C v C .v r/v C rq D g; @t

div v D 0;

and

11.2 Determining Nodes

257

with the same homogeneous Dirichlet boundary conditions; p; q and f ; g are the corresponding pressures and external forces, respectively. We assume that the forcing terms f and g, f ; g 2 L1 .0; 1I H/, have the same asymptotic behavior as time goes to infinity, Z ˝

jf .x; t/ g.x; t/j2 dx ! 0

as

t ! 1:

Let ˝ be covered by N identical squares Q1 ; : : : ; QN (for example, by a regular mesh) and let each square Qi contain exactly one measurement point xi , i D 1; : : : ; N. We thus have a set of N measurement points E D fx1 ; : : : ; xN g. Let max ju.xj ; t/ v.xj ; t/j ! 0

jD1;:::;N

as

t ! 1:

(11.9)

The set E is then called a set of determining nodes if (11.9) implies Z ˝

ju.x; t/ v.x; t/j2 dx ! 0

as

t ! 1:

We shall show that if (11.9) holds for some N then also Z jru.x; t/ rv.x; t/j2 dx ! 0 as t ! 1: ˝

Denote .w/ D max jw.xj /j: 1jN

To prove the existence of a finite number of determining nodes we shall use the generalized Gronwall lemma (cf. Lemma 3.23) as well as inequality (11.10) of the following lemma. Lemma 11.1. Let the domain ˝ be covered by N identical squares. Consider the set E D fx1 ; : : : ; xN g of points in ˝, distributed one in each square. Then for each vector field w in D.A/ the following inequalities hold, c c .w/2 C 2 2 jAwj2 ; 1 1 N c kwk2 cN.w/2 C jAwj2 ; 1 N c jAwj2 ; kwk2L1 .˝/ cN.w/2 C 1 N jwj2

where the constant c depends only on the shape of the domain ˝.

(11.10)

258


A proof of the lemma can be found in [99]. Theorem 11.3. Let the domain ˝ be covered by N identical squares. Consider the set E D fx1 ; : : : ; xN g of points in ˝, distributed one in each square. Let f and g be two forcing terms in L1 .0; 1I H/ that satisfy Z ˝

jf .x; t/ g.x; t/j2 dx ! 0

as

t ! 1;

and let F D lim sup jf .t/j D lim sup jg.t/j as t!1

t ! 1:

t!1

Then there exists a constant C D C.F; ; ˝/ such that if N C D C.F; ; ˝/; then E is a set of determining nodes in the sense defined above for the twodimensional Navier–Stokes equations with homogeneous Dirichlet boundary conditions. Proof. Let w D u v. We know that .w.t// ! 1 as t ! 1. We have to show that jw.t/j ! 0 as well. Actually, we will show that kw.t/k ! 0 as

t ! 1:

We have, from the Navier–Stokes equations, dw.t/ C Aw.t/ C B.w.t/; u.t// C B.v.t/; w.t// D f .t/ g.t/: dt Taking the inner product with Aw.t/ in H we obtain 1d kw.t/k2 C jAw.t/j2 C b.w.t/; u.t/; Aw.t// C b.v.t/; w.t/; Aw.t// 2 dt D .f .t/ g.t/; Aw.t//:

(11.11)

By C we shall denote a generic constant that might depend on ˝; f ; g; , but not on the initial conditions u0 ; v0 . By c we denote constants that might depend only on ˝. We estimate the nonlinear terms in the following way: c jAwj2 C 3 kuk4 jwj2 ; 6 c 1=2 1=2 2 jb.v; w; Aw/j c1 jvj jAvj kwkjAwj jAwj C jvjjAvjkwk2 : 6 jb.w; u; Aw/j c1 jwj1=2 kukjAwj3=2

11.2 Determining Nodes

259

Exercise 11.5. Prove the above two inequalities by using the following Agmon inequality: kuk1 c2 juj1=2 jAuj1=2

for

u 2 D.A/:

(11.12)

The right-hand side of (11.11) is estimated as j.f g; Aw/j

3 jAuj2 C jf gj2 : 6 2

Thus, (11.11) yields d c c 3 kwk2 C jAwj2 3 kuk4 jwj2 C jvjjAvjkwk2 C jf gj2 : dt

(11.13)

From (11.10) we have jAwj2 c1 1 Nkwk2 1 N 2 .w/2 : Using the Poincaré inequality in the first term on the right-hand side of (11.13) we get, in the end, d c c 2 1 4 kwk C c 1 N 3 kuk jvjjAvj kwk2 dt 1

3 jf gj2 C 1 N 2 .w/2 :

This inequality can be written in the form d C ˛ ˇ dt

where

D kw.t/k2 :

(11.14)

We see that ˇ.t/ ! 0 as

t ! 1;

and hence 1 t!1 T

Z

tCT

lim

ˇ C . /d D 0:

(11.15)

t ! jv.t/j

(11.16)

t

As the functions t ! ku.t/k

and

260


are bounded on Œt0 ; 1/ for t0 > 0, and 1 T

Z

tCT t

jAv./jd C.jgj2L1 .t;tCTIH/ ; ; ˝/;

(11.17)

we deduce that 1 lim inf t!1 T

Z

tCT

˛./d c1 1 N C.f ; g; ; ˝/:

t

For N sufficiently large we have 1 lim inf t!1 T

Z

tCT

˛./d > 0:

(11.18)

˛ . /d < 1:

(11.19)

t

Also lim sup t!1

1 T

Z

tCT

t

In view of (11.14) together with (11.15), (11.18), and (11.19) we deduce that .t/ D kw.t/k ! 0

as

t ! 1:

Hence, the set E is a set of determining nodes.

t u

Exercise 11.6. Prove (11.17) from 1d kv.t/k2 C jAv.t/j2 C b.v.t/; v.t/; Av.t// D .g.t/; Av.t//; 2 dt knowing that the functions in (11.16) are bounded.

11.3 Pullback Attractors for Asymptotically Compact Non-autonomous Dynamical Systems Let ˝ be a nonempty set and let f t gt2R be a family of mappings t W ˝ ! ˝ satisfying: 1. 0 ! D ! for all 2. t . !/ D tC !

! 2 ˝, for all ! 2 ˝

and all

t; 2 R:

The mappings t are called the shift operators. Let moreover X be a metric space with the distance d.; /; and let be a cocycle on X, i.e., a mapping W RC ˝ X ! X; which satisfies:

11.3 Pullback Attractors for Asymptotically Compact Non-autonomous. . .

(a) .0; !; x/ D x for all .!; x/ 2 ˝ X, (b) .tC; !; x/ D .t; !; .; !; x// for all

261

t; 2 RC and .!; x/ 2 ˝ X.

The -cocycle is said to be continuous if for all .t; !/ 2 RC ˝, the mapping .t; !; / W X ! X is continuous. The sets parameterized by the index ! 2 ˝ will be called the parameterized sets b D fD.!/g!2˝ , while their particular case, the sets parameterized by and denoted D time t 2 R will be called the non-autonomous sets and denoted D D fD.t/gt2R : This different notation stresses the special case ˝ D R and t s D sCt. Let P.X/ denote the family of all nonempty subsets of X, and S the class of all parameterized sets b D fD.!/g!2˝ such that D.!/ 2 P.X/ for all ! 2 ˝. We say that the fibers D b are nonempty. D.!/ of the parameterized set D We consider a given nonempty subclass D S : This class will be called an attraction universe. Definition 11.2. The -cocycle is said to be pullback D-asymptotically compact b 2 D, and any sequences tn ! C1, xn 2 .D-a.c./ if for any ! 2 ˝, any D D. tn !/; the sequence .tn ; tn !; xn / has a convergent subsequence. b D fB.!/g!2˝ 2 S is said to be pullback Definition 11.3. A parameterized set B b b 0 such that D-absorbing if for each ! 2 ˝ and D 2 D, there exists t0 .!; D/ .t; t !; D. t !// B.!/

for all

b t t0 .!; D/:

We denote by distX .C1 ; C2 / the Hausdorff semi-distance between C1 and C2 , defined as distX .C1 ; C2 / D sup inf d.x; y/ x2C1 y2C2

for

C1 ; C2 X:

b D fC.!/g!2˝ 2 S is said to be pullback Definition 11.4. A parameterized set C D-attracting if lim distX ..t; t !; D. t !//; C.!// D 0 for all

t!C1

b 2 D; ! 2 ˝: D

b D fA.!/g!2˝ 2 S is said to be a global Definition 11.5. A parameterized set A pullback D-attractor if it satisfies (i) A.!/ is compact for any ! 2 ˝; b is pullback D-attracting, (ii) A b is invariant, i.e., (iii) A .t; !; A.!// D A. t !/

for any

.t; !/ 2 RC ˝:

Remark 11.2. Observe that Definition 11.5 does not guarantee the uniqueness of a pullback D-attractor (see [36] for a discussion on this point). In order to ensure the

262


uniqueness one needs to impose additional conditions as, for instance, the condition that the attractor belongs to the same attraction universe D, or some kind of minimality. However, we do not include this stronger assumptions in the definition since, as we will show in Theorem 11.4, under very general hypotheses, it is possible to ensure the existence of a global pullback D-attractor which is minimal in an appropriate sense. b 2 S and ! 2 ˝; we define the !-limit set of D b at Definition 11.6. For each D ! as ! \ [ b .D; !/ D .t; t !; D. t !// : s0

ts

b !/ is a closed subset of X that can be possibly empty. It is easy Obviously, .D; b !/ if and only if there exist a to see that for each y 2 X; one has that y 2 .D; sequence tn ! C1 and a sequence xn 2 D. tn !/ such that lim d..tn ; tn !; xn /; y/ D 0:

tn !C1

Our first aim is to prove the following result on the existence of global pullback D-attractor. Theorem 11.4. Suppose the -cocycle is continuous and pullback Db 2 D which is pullback D-absorbing. asymptotically compact, and there exists B b Then, the parameterized set A defined by b !/ A.!/ D .B;

for

! 2 ˝;

b 2 S is a is a global pullback D-attractor which is minimal in the sense that if C parameterized set such that C.!/ is closed and lim distX ..t; t !; B. t !//; C.!// D 0;

t!C1

then A.!/ C.!/. In order to prove Theorem 11.4, we first need the following results. b 2 S is a pullback D-absorbing parameterized set, then Proposition 11.1. If B b !/ .B; b !/ .D;

for all

b2 D D

and

! 2 ˝:

b 2 D, then If in addition B b !/ .B; b !/ B.!/ .D;

b2 D for all D

and

! 2 ˝:


263

b 2 D, ! 2 ˝; and y 2 .D; b !/. There exist a sequence tn ! Proof. Let us fix D C1 and a sequence xn 2 D. tn !/ such that .tn ; tn !; xn / ! y:

(11.20)

b is pullback D-absorbing, for each integer k 1 there exists an index nk such As B that tnk k and .tnk k; .tnk k/ . k !/; D. .tnk k/ . k !/// B. k !/:

(11.21)

In particular, as xnk 2 D. tnk !/ D D. .tnk k/ . k !//; we obtain from (11.21) yk D .tnk k; tnk !; xnk / 2 B. k !/: But then .tnk ; tnk !; xnk / D ..tnk k/ C k; tnk !; xnk / D .k; k !; .tnk k; tnk !; xnk // D .k; k !; yk /; and consequently, by (11.20), .k; k !; yk / ! y with

yk 2 B. k !/:

b !/. Thus, y belongs to .B; b !/, then there exist a sequence n ! Finally, observe that if z belongs to .B; b is C1 and a sequence wn 2 B. n !/ such that .n ; n !; wn / ! z: But, as B b pullback D-absorbing, there exists t0 .B; !/ 0 such that .n ; n !; wn / 2 B.!/ b !/; and consequently z belongs to B.!/: t u for all n t0 .B; Remark 11.3. As a straightforward consequence of Proposition 11.1, we have that b 2 D is pullback D-absorbing, then if B [

b !/ D .B; b !/; .D;

b D2D so that, under the assumptions in Theorem 11.4, we have b !/ D A.!/ D .B;

[ b D2D

b !/ .D;

for ! 2 ˝:

264


b 2 D and Proposition 11.2. If is pullback D-asymptotically compact, then for D b !/ is nonempty, compact, and ! 2 ˝; the set .D; b !// D 0: lim distX ..t; t !; D. t !//; .D;

t!C1

(11.22)

b 2 D and ! 2 ˝. If we consider a sequence tn ! C1, and Proof. Let us fix D a sequence xn 2 D. tn !/; then from the sequence .tn ; tn !; xn / we can extract a convergent subsequence .t ; t !; x / ! y. By its construction, y belongs to b !/, and thus this set is nonempty. .D; b !/ is closed, and consequently in order to prove its We know that .D; b !/, we compactness it is enough to see that for any given sequence fyn g .D; b can extract a convergent subsequence. First, observe that as yn 2 .D; !/, there exist tn n and xn 2 D. tn !/ such that d..tn ; tn !; xn /; yn /

1 : n

(11.23)

As is pullback D-a.c., from the sequence .tn ; tn !; xn / we can extract a convergent subsequence .t ; t !; x / ! y, and consequently, from (11.23), we also obtain y ! y: Finally, if (11.22) does not hold, there exist " > 0; a sequence tn ! C1, and a sequence xn 2 D. tn !/; such that d..tn ; tn !; xn /; y/ " for all

b !/: y 2 .D;

(11.24)

But from the sequence .tn ; tn !; xn / we can extract a convergent subsequence b !/; which contradicts (11.24). t u .t ; t !; x / ! y 2 .D; Proposition 11.3. If the -cocycle is continuous and pullback D-asymptotically b 2 D; one has compact, then for any .t; !/ 2 RC ˝ and any D b !// D .D; b t !/: .t; !; .D; b 2 D be fixed, and let y 2 .D; b !/. We show Proof. Let .t; !/ 2 RC ˝ and D b that .t; !; y/ 2 .D; t !/; and consequently we obtain b !// .D; b t !/: .t; !; .D; We already know that there exist a sequence tn ! C1 and a sequence xn 2 D. tn !/ such that .tn ; tn !; xn / ! y:


265

But .t; !; .tn ; tn !; xn // D .t C tn ; .tCtn / . t !/; xn /; and consequently, as is continuous, .t C tn ; .tCtn / . t !/; xn / ! .t; !; y/: Since we have t C tn ! C1 and xn 2 D. tn !/ D D. .tCtn / . t !//; it follows that b t !/: .t; !; y/ 2 .D; b t !/ .t; !; .D; b !//: For y 2 .D; b t !/ there exist a Now, we prove .D; sequence tn ! C1 and a sequence xn 2 D. tn . t !// such that .tn ; tn . t !/; xn / ! y:

(11.25)

If tn t; we have .tn ; tn . t !/; xn / D .t C .tn t/; .tn t/ !; xn / D .t; !; .tn t; .tn t/ !; xn //:

(11.26)

But, as is pullback D-a.c., tn t ! C1; and xn 2 D. tn . t !// D D. .tn t/ !/, one can ensure that there exists a subsequence f.t ; x /g f.tn ; xn /g such that b !/; .t t; .t t/ !; x / ! z 2 .D; and then, by (11.26) and (11.25), b !//: u y D .t; !; z/ 2 .t; !; .D; t Now, we can prove the main Theorem. Proof (of Theorem 11.4). The compactness of each S A.!/ follows from Proposib !/ A.!/; we tion 11.2. Also, by this proposition and the fact that b .D; D2D obtain lim distX ..t; t !; D. t !//; A.!// D 0

t!C1

for any

b 2 D: D

Now, observe that, by Proposition 11.3, A. t !/ D

[ b D2D

b t !/ D .D;

[ b D2D

b !// D .t; !; .t; !; .D;

[ b D2D

b !//; .D;

266


and, as is continuous and .t; !;

S b !/ is compact, .D; b D2D

[

b !// D .t; !; .D;

b D2D

[

b !//: .D;

b D2D

Consequently, we have the invariance property .t; !; A.!// D A. t !/: Now, let b C 2 S be a parameterized set such that C.!/ is closed and lim distX ..t; t !; B. t !//; C.!// D 0:

t!C1

Let y be an element of A.!/, then y D lim .tn ; tn !; xn /; n!1

for some sequences tn ! C1, xn 2 B. tn !/; and consequently y 2 C.!/ D C.!/: Thus, A.!/ C.!/. u t Remark 11.4. Observe that if in Theorem 11.4 we assume that B.!/ is closed for b0 2 S , D b2 D all ! 2 ˝; and the attraction universe D is inclusion-closed (i.e., if D b0 2 D), then it follows that the pullback and D0 .!/ D.!/ for all ! 2 ˝, then D b belongs to D, and hence it is the unique pullback D-attractor which D-attractor A belongs to D. This situation appears very often in applications, in particular, in our example in Sect. 11.4. Remark 11.5. Note that we do not assume any structure (neither measurable nor topological) on the set of parameters ˝. Consequently, this result can be applied to both non-autonomous and random dynamical systems. Although the cocycle formalism is very useful, in particular, in the case of random dynamical systems or non-autonomous problems, we state below our main result (namely Theorem 11.4) in the language of evolutionary processes (see [225]) and then apply it in the theory of Navier–Stokes equations. Let us consider an evolutionary process (also called a two-parameter semigroup or just a process) U on X, i.e., a family fU.t; /g1< t
for all

r t:

(11.27)

The attraction universe D will now be a nonempty class of non-autonomous sets D D fD.t/gt2R such that D.t/ 2 P.X/ for all t 2 R. Definition 11.7. The process U is said to be pullback D-asymptotically compact if for any t 2 R, any D 2 D; any sequence n ! 1; and any sequence xn 2 D.n /, the sequence fU.t; n /xn g is relatively compact in X.


267

Fig. 11.1 Illustration of a pullback D-absorbing property. The sets D.ti / become larger as ti tend to 1 but still at time t all trajectories originating from those sets are in B.t/

Definition 11.8. It is said that B D fB.t/gt2R 2 D is pullback D-absorbing for the process U if for any t 2 R and any D 2 D, there exists a 0 .t; D/ t such that U.t; /D. / B.t/

for all

0 .t; D/:

The notion of pullback Dabsorbing set is illustrated in Fig. 11.1. We have the following result. Theorem 11.5. Suppose that the process U is pullback D-asymptotically compact and that B 2 D is a non-autonomous D-absorbing set for U. Then, the non-autonomous set A D fA.t/gt2R with nonempty fibers A.t/ 2 P.X/ for t 2 R defined by A.t/ D .B; t/

for t 2 R;

where for each D D fD.t/gt2R 2 D .D; t/ D

\ [ st

! U.t; /D. / ;

s

has the following properties: (i) the sets A.t/ are compact for t 2 R, (ii) the non-autonomous set A is pullback D-attracting, i.e., lim distX .U.t; /D. /; A.t// D 0 for all

!1

D 2 D;

268


(iii) the non-autonomous set A is invariant, i.e., U.t; /A. / D A.t/ for

1 < t < C1;

(iv) the sets A.t/ are given by A.t/ D

[

.D; t/

for t 2 R:

D2D

The non-autonomous set A, termed the global pullback D-attractor for the process U, is minimal in the sense that if C D fC.t/gt2R with C.t/ 2 P.X/ is a non-autonomous set such that C.t/ is closed and lim distX .U.t; /B. /; C.t// D 0;

!1

then A.t/ C.t/. Proof. It is enough to denote ˝ D R, and t D C t; and to apply Theorem 11.4 to the non-autonomous dynamical system defined by .t; ; x/ D U.t C ; /x: u t Remark 11.6. The comments contained in Remark 11.4 also hold true for evolutionary processes.

11.4 Application to Two-Dimensional Navier–Stokes Equations in Unbounded Domains Let ˝ R2 be an open set, not necessarily bounded, with the boundary @˝, and suppose that ˝ satisfies the Poincaré inequality, i.e., there exists a constant 1 > 0 such that Z Z 1 2 dx jrj2 dx for all 2 H01 .˝/: (11.28) ˝

˝

Consider the following two-dimensional Navier–Stokes problem: 8 2 X ˆ @u @u ˆ ˆ u C ui D f .t/ rp in ˝ .; C1/; ˆ ˆ @xi < @t iD1 div u D 0 ˆ ˆ ˆ ˆ uD0 ˆ : u.; x/ D u0 .x/

in ˝ .; C1/; on @˝ .; C1/; for x 2 ˝:

11.4 Application to Two-Dimensional Navier–Stokes Equations. . .

269

To set our problem in the abstract framework, we consider the following usual abstract space o n 2 VQ D u 2 C01 .˝/ W div u D 0 : We define H as the closure of VQ in .L2 .˝//2 with the norm jj ; and the inner product .; / where for u; v 2 .L2 .˝//2 we set .u; v/ D

2 Z X jD1

˝

uj .x/vj .x/ dx:

Moreover we define V as the closure of VQ in .H01 .˝//2 with the norm k k; and the associated scalar product ..; //; where for u; v 2 .H01 .˝//2 we have 2 Z X @uj @vj ..u; v// D dx: @xi @xi i;jD1 ˝

It follows that V H H 0 V 0 ; where the injections are continuous and dense. Finally, we will use kkV 0 for the norm in V 0 and h; i for the duality pairing between V and V 0 : Consider the trilinear form b on V V V given by b.u; v; w/ D

2 Z X i;jD1 ˝

ui

@vj wj dx @xi

for u; v; w 2 V:

Define B W V V ! V 0 by hB.u; v/; wi D b.u; v; w/

for

u; v; w 2 V;

and denote B.u/ D B.u; u/: 2 .RI V 0 /. For each 2 R we consider the problem Assume now that u0 2 H, f 2 Lloc

8 ˆ u 2 L2 .; TI V/ \ L1 .; TI H/ for all T > ; ˆ ˆ ˆ < d .u.t/; v/ C ..u.t/; v// C hB.u.t//; vi D hf .t/; vi for all dt ˆ ˆ in the sense of scalar distributions on .; C1/; ˆ ˆ : u. / D u0 :

v2V

(11.29)

270


It follows from the results of Chap. 7, cf. Theorem 7.2 and Remark 7.3, that problem (11.29) has a unique solution, u.I ; u0 /, that moreover belongs to C.Œ; TI H/ for all T . Define U.t; /u0 D u.tI ; u0 /

for

t

and

u0 2 H:

(11.30)

From the uniqueness of solution to problem (11.29), it follows that U.t; /u0 D U.t; r/U.r; /u0

rt

for all

and

u0 2 H:

(11.31)

One can prove (cf. Chap. 7) that for all t; the mapping U.; t/ W H ! H defined by (11.30) is continuous. Consequently, the family fU.t; /g t defined by (11.30) is a process U in H. Moreover, it can be proved that U is weakly continuous, and more exactly the following result holds true. Its proof is identical to the proof of Lemma 8:1 in [164], and so we omit it. Proposition 11.4. Let fu0n g H be a sequence converging weakly in H to an element u0 2 H: Then U.t; /u0n ! U.t; /u0 U.; /u0n ! U.; /u0

weakly in

weakly in

H

for all

L2 .; TI V/

t;

for all

< T:

(11.32) (11.33)

From now on, we denote D 1 :

(11.34)

Let R be the set of all functions r W R ! .0; C1/ such that lim et r2 .t/ D 0;

(11.35)

t!1

and denote by D the class of all non-autonomous sets D D fD.t/gt2R such that D.t/ 2 P.H/ for t 2 R, for which there exists rD 2 R such that D.t/ B.0; rD .t//; where B.0; rD .t// denotes the closed ball in H centered at zero with radius rD .t/: Now, we can prove the following result. 2 Theorem 11.6. Suppose that f 2 Lloc .RI V 0 / is such that

Z

t

1

e kf ./k2V 0 d < C1

for all

t 2 R:

(11.36)


271

Then, there exists a unique global pullback D -attractor for the process U defined by (11.30). Proof. Let 2 R and u0 2 H be fixed, and denote u.t/ D u.tI ; u0 / D U.t; /u0

for all

t :

Taking into account that b.u; v; v/ D 0; it follows d t e ju.t/j2 C 2et ku.t/k2 D et ju.t/j2 C 2et hf .t/; u.t/i dt 0 in C01 .; C1/ ; and consequently, by (11.28), 1 e ju.t/j e ju0 j C t

2

2

Z

t

e kf ./k2V 0 d

for all t:

(11.37)

Let D 2 D be given. From (11.37), we easily obtain jU.t; /u0 j2 e.t / rD2 . / C

et

Z

t 1

e kf ./k2V 0 d

(11.38)

for all u0 2 D. /; and all t . Denote by R .t/ the nonnegative number given for each t 2 R by .R .t//2 D

2et

Z

t

1

e kf ./k2V 0 d;

and consider the non-autonomous set B D fB .t/gt2R of closed balls in H defined by B .t/ D fv 2 H W jvj R .t/g:

(11.39)

It is straightforward to check that B 2 D , and moreover, by (11.35) and (11.38), the family B is pullback D -absorbing for the process U. According to Theorem 11.5 and Remark 11.6, to finish the proof of the theorem we only have to prove that U is pullback D -asymptotically compact. Let D 2 D , a sequence n ! 1, a sequence u0n 2 D.n / and t 2 R, be fixed. We must prove that from the sequence fU.t; n /u0n g we can extract a subsequence that converges in H. As the non-autonomous set B is pullback D -absorbing, for each integer k 0 there exists a D .k/ t k such that U.t k; /D. / B .t k/

for all

D .k/:

(11.40)

272


It is not difficult to conclude from (11.40), by a diagonal procedure, the existence of a subsequence f.n0 ; u0n0 /g f.n ; u0n /g; and a sequence fwk g1 kD0 H; such that for all k D 0; 1; 2; : : : we have wk 2 B .t k/ and U.t k; n0 /u0n0 ! wk

weakly in H:

(11.41)

Observe that, by Proposition 11.4, w0 D weak 0lim U.t; n0 /u0n0 n !1

D weak 0lim U.t; t k/U.t k; n0 /u0n0 n !1

D U.t; t k/.weak 0lim U.t k; n0 /u0n0 /; n !1

i.e., U.t; t k/wk D w0

for all k D 0; 1; 2; : : : :

(11.42)

Then, by the weak lower semicontinuity of the norm, inf jU.t; n0 /u0n0 j: jw0 j lim 0

(11.43)

lim sup jU.t; n0 /u0n0 j jw0 j;

(11.44)

n !1

If we now prove that also n0 !1

then we will have lim jU.t; n0 /u0n0 j D jw0 j;

n0 !1

and this, together with the weak convergence, will imply the strong convergence in H of U.t; n0 /u0n0 to w0 . In order to prove (11.44), we consider the Hilbert norm in V given by Œu2 D kuk2

2 juj ; 2

which by (11.28) is equivalent to the norm kuk in V: It is immediate that for all t and all u0 2 H; (11.45) jU.t; /u0 j2 D ju0 j2 e.t/ Z t C2 e.t/ hf ./; U.; /u0 i ŒU.; /u0 2 d;


273

and, thus, for all k 0 and all n0 t k; jU.t; n0 /u0n0 j2 D jU.t; t k/U.t k; n0 /u0n0 j2

(11.46)

D ek jU.t k; n0 /u0n0 j2 Z t C2 e.t/ hf ./; U.; t k/U.t k; n0 /u0n0 i d Z 2

tk t

e.t/ ŒU.; t k/U.t k; t n0 /u0n0 2 d:

tk

As, by (11.40), U.t k; n0 /u0n0 2 B .t k/

for all

n0 D .k/

and

k 0;

we have lim sup ek jU.t k; n0 /u0n0 j2 ek R2 .t k/ n0 !1

for all

k 0:

(11.47)

On the other hand, as U.t k; n0 /u0n0 ! wk weakly in H, from Proposition 11.4 we have weakly in L2 .t k; tI V/:

U.; t k/U.t k; n0 /u0n0 ! U.; t k/wk

(11.48)

Taking into account that, in particular, e.t/ f ./ 2 L2 .t k; tI V 0 /; we obtain from (11.48), Z t e.t/ hf ./; U.; t k/U.t k; n0 /u0n0 i d (11.49) lim 0 n !1 tk

Z D

t

e.t/ hf ./; U.; t k/wk i d:

tk

1=2

R t

defines a norm in L2 .t k; tI V/ which is Moreover, as tk e.t/ Œv./2 d equivalent to the usual one, we also obtain from (11.48), Z t e.t/ ŒU.; t k/wk /2 d (11.50) tk

Z lim inf 0 n !1

t

e.t/ ŒU.; t k/U.t k; n0 /u0n0 2 d:

tk

Then, from (11.46), (11.47), (11.49), and (11.50), we easily obtain lim sup jU.t; n0 /u0n0 j2 n0 !1

ek R2 .t

(11.51) Z

k/ C 2

t tk

e.t/ hf ./; U.; t k/wk i ŒU.; t k/wk /2 d:

274


Now, from (11.42) and (11.45), jw0 j2 D jU.t; t k/wk j2 Z 2 k D jwk j e C2

(11.52) t

e.t/ hf ./; U.; t k/wk i ŒU.; t k/wk 2 d:

tk

From (11.51) and (11.53) we have lim sup jU.t; n0 /u0n0 j2 ek R2 .t k/ C jw0 j2 jwk j2 ek n0 !1

ek R2 .t k/ C jw0 j2 ; and thus, taking into account that ek R2 .t k/ D

2et

Z

tk

1

e kf ./k2V 0 d ! 0;

when k ! C1, we easily obtain (11.44) from the last inequality.

t u

In Chap. 13 we consider, in a similar framework, a problem from the theory of lubrication and prove the finite dimensionality of the pullback attractor.

11.5 Comments and Bibliographical Notes The presentation in Sects. 11.1 and 11.2 follows [99]. Estimates on the number of determining modes and determining nodes can be found in [122]. We note here that another possibility of finite dimensional reduction of infinite dimensional dynamical systems is through the study of inertial manifolds [77] or of invariant manifolds [54]. For Navier–Stokes equations the approach by inertial manifolds leads to the socalled approximate inertial manifolds studied in [224], existence of inertial manifold for two-dimensional Navier–Stokes equations is still unresolved [103]. The results of Sects. 11.3 and 11.4 come from [45], where the notion of an asymptotically compact dynamical system as in Definition 11.2 was introduced. The notion of pullback attractor is a direct generalization of that of global attractor to the non-autonomous case. The first attempts to extend the notion of global attractor to the non-autonomous case led to the concept of the so-called uniform attractor (see [62]). The conditions ensuring the existence of the uniform attractor parallel those for autonomous systems. In this theory non-autonomous systems are lifted in [225] to autonomous ones by expanding the phase space. Then, the existence of uniform attractors relies on some compactness property of the solution operator associated to the resulting system. One disadvantage of uniform attractors is that they do not need to be invariant unlike the global attractor for autonomous systems. Moreover, they demand rather strong conditions on the time-dependent data.


275

At the same time, the theory of pullback (or cocycle) attractors has been developed for both the non-autonomous and random dynamical systems (see [80, 137, 152, 206, 232]), and has shown to be very useful in the understanding of the dynamics of non-autonomous dynamical systems. In this case, the concept of pullback (or cocycle or non-autonomous) attractor provides a time-dependent family of compact sets which attracts families of sets in a certain universe (e.g., the bounded sets in the phase space) and satisfying an invariance property, what seems to be a natural set of conditions to be satisfied for an appropriate extension of the autonomous concept of attractor. Moreover, this cocycle formulation allows to handle more general time-dependent terms in the models—not only the periodic, quasi-periodic, or almost periodic ones (see, for instance, [38], and [43] for nonautonomous models containing hereditary characteristics). Note that, however, the notion of pullback attractor has its limitations, namely it does not always capture the asymptotic behavior of the non-autonomous system as time advances to C1, see [140]. To counteract this limitation the synthesis of the approach by pullback attractors with the one by uniform attractors has been presented in [19, 53, 136] In order to prove the existence of the attractor (in both the autonomous and non-autonomous cases) the simplest, and therefore the strongest, assumption is the compactness of the solution operator associated with the system, which is usually available for parabolic systems in bounded domains. However, this kind of compactness does not hold in general for parabolic equations in unbounded domains and second order in time equations on either bounded or unbounded domains. Instead we often have some kind of asymptotic compactness. In this situation, there are several approaches to prove the existence of the global (or non-autonomous) attractor. Roughly speaking, the first one ensures the existence of the global (resp. non-autonomous) attractor whenever a compact attracting set (resp. a family of compact attracting sets) exists. The second method consists in decomposing the solution operator (resp. the cocycle or two-parameter semigroup) into two parts: a compact part and another one which decays to zero as time goes to infinity. However, as it is not always easy to find such decomposition, one can use a third approach which is based on the use of the energy equations which are in direct connection with the concept of asymptotic compactness. This third method has been used in [164] (and also [181]) to extend to the non-autonomous situation the corresponding one in the autonomous framework (see [201] and also [11]), but related to uniform asymptotic compactness. The method used in this chapter is a generalization of the latter to the theory of pullback attractors. More information on non-autonomous infinite dimensional dynamical systems can be found in monographs [53, 136] and the review article [10], cf. also [208]. For the random case, see, e.g., [30, 31, 39, 95, 135, 153]. Two-dimensional problems with delay have been studied in [37, 170].

12

Pullback Attractors and Statistical Solutions

This chapter is devoted to constructions of invariant measures and statistical solutions for non-autonomous Navier–Stokes equations in bounded and certain unbounded domains in R2 . After introducing some basic notions and results concerning attractors in the context of the Navier–Stokes equations, we construct the family of probability measures f t gt2R and prove the relations t .E/ D .U.t; /1 E/ for t; 2 R, t and Borel sets E in H. Then we prove the Liouville and energy equations. Finally, we consider statistical solutions of the Navier–Stokes equations supported on the pullback attractor.

12.1 Pullback Attractors and Two-Dimensional Navier–Stokes Equations We begin this section by recalling the notion of pullback attractor introduced in Chap. 11 in a more general context of cocycles. Let us consider an evolutionary process U (a process U—for short) on a metric space X, i.e., a family fU.t; /g1< t
for all

r t:

Let D be an attraction universe, i.e., a nonempty family of non-autonomous sets D D fD.t/gt2R such that D.t/ 2 P.X/ for all t 2 R, where P.X/ denotes the family of all nonempty subsets of X.


277

278

12 Pullback Attractors and Statistical Solutions

Definition 12.1. A process U.; / is said to be pullback D-asymptotically compact if for any t 2 R, any D 2 D; any sequence n ! 1; and any sequence xn 2 D.n /, the sequence fU.t; n /xn g is relatively compact in X. Definition 12.2. It is said that B 2 D is pullback D-absorbing for the process U.; / if for any t 2 R and any D 2 D, there exists a 0 .t; D/ t such that U.t; /D. / B.t/ for all

0 .t; D/:

Definition 12.3. A non-autonomous set A D fA.t/gt2R with A.t/ 2 P.X/ for all t 2 R is said to be a pullback D-attractor for the process U.; / if (i) A.t/ is compact for all t 2 R, (ii) A is pullback D-attracting, i.e., lim distX .U.t; /D. /; A.t// D 0 for all

!1

D2D

and all

t 2 R;

(iii) A is invariant, i.e., U.t; /A. / D A.t/ for 1 < t < C1. We consider the initial and boundary value problem for the Navier–Stokes equations studied already in Chap. 11. Namely, let ˝ R2 be an open, bounded or unbounded, domain with smooth boundary @˝ such that there exists a constant 1 > 0 for which Z Z 2 dx jrj2 dx for all 2 H01 .˝/: 1 ˝

˝

Consider the following problem: 8 2 X ˆ @u @u ˆ ˆ u C ui D f .t/ rp in ˝ .; C1/; ˆ ˆ @xi < @t iD1 div u D 0 ˆ ˆ ˆ ˆ uD0 ˆ : u.; x/ D u0 .x/;

in ˝ .; C1/; on @˝ .; C1/; for x 2 ˝:

This problem is set in a suitable abstract framework by considering the usual function space n o 2 VQ D u 2 C01 .˝/ W div u D 0 : The space H is defined as the closure of VQ in .L2 .˝//2 with the norm jj, and the associated scalar product .; / given by .u; v/ D

2 Z X jD1

˝

uj .x/vj .x/ dx

for

u; v 2 .L2 .˝//2 :

12.1 Pullback Attractors and Two-Dimensional Navier–Stokes Equations

279

The space V is defined as the closure of VQ in .H01 .˝//2 with the norm kk ; and the associated scalar product ..; //; given by .ru; rv/ D ..u; v// D

2 Z X @uj @vj dx @xi @xi i;jD1 ˝

u; v 2 .H01 .˝//2 :

for

It follows that V H H 0 V 0 ; where the injections are continuous and dense. Finally, we will use kkV 0 for the norm in V 0 and h; i for the duality pairing between V and V 0 : Let A W V ! V 0 be given by hAu; vi D ..u; v// for u; v 2 V, and consider the trilinear form b on V V V given by b.u; v; w/ D

2 Z X

ui

i;jD1 ˝

@vj wj dx @xi

for u; v; w 2 V:

Define B W V V ! V 0 by hB.u; v/; wi D b.u; v; w/; for u; v; w 2 V; and denote 2 B.u/ D B.u; u/: Assume now that u0 2 H, f 2 Lloc .RI V 0 /. For each 2 R we consider the problem 8 ˆ u 2 L2 .; TI V/ \ L1 .; TI H/ for all T > ; ˆ ˆ ˆ < d .u.t/; v/ C ..u.t/; v// C hB.u.t//; vi D hf .t/; vi for all dt ˆ ˆ in the sense of scalar distributions on .; C1/; ˆ ˆ : u. / D u0 :

v2V

(12.1) It follows from the results of Chap. 7, cf. Theorem 7.2 and Remark 7.3, that problem (12.1) has a unique solution, u.I ; u0 /, that moreover belongs to C.Œ; TI H/ for all T . Now we formulate a result about the existence of the pullback attractor for the above Navier–Stokes problem. We define the evolutionary process associated to (12.1) as follows. Let us consider the unique solution u.I ; u0 / to (12.1). We set U.t; /u0 D u.tI ; u0 /

for t

and

u0 2 H:

(12.2)

By the uniqueness of the solution we have U.t; /u0 D U.t; r/U.r; /u0

for all

rt

and

u0 2 H:

For all t; the mapping U.; t/ W H ! H defined by (12.2) is continuous (cf. Theorem 7.2 and Remark 7.3). Thus, the family fU.t; /g t defined by (12.2) is a process U.; / in H. Let D 1 and R be the set of all functions r W R ! .0; C1/ such that lim et r2 .t/ D 0;

t!1

280


and denote by D the class of all non-autonomous sets D D fD.t/gt2R with D.t/ 2 P.H/ for all t 2 R such that D.t/ B.0; rD .t//; for some rD 2 R ; where B.0; rD .t// denotes the closed ball in H centered at zero with radius rD .t/. Then we have (see Chap. 11). 2 Theorem 12.1. Suppose that f 2 Lloc .RI V 0 / is such that

Z

t

1

e kf ./k2V 0 d < C1

for all

t 2 R:

(12.3)

Then, the process U.; / associated with problem (12.1) is D -asymptotically compact and there exists B 2 D which is pullback D -absorbing for U.t; /. In consequence, there exists a unique pullback D -attractor A D fA.t/gt2R belonging to D for the process U.; / in H. The choice of the attraction universe D comes here naturally from the first energy inequality (cf. Chap. 11) ju.t/j2 e.t / ju. /j2 C

1 t e

Z

t

e kf ./k2V 0 d

for t:

If u. / 2 D. / B.0; rD . //, rD 2 R , then the first term on the right-hand side converges to zero as ! 1, and 2 ju.t/j et 2

Z

t

1

e kf ./k2V 0 d R.t/

for all

0 .t; D/:

(12.4)

Thus U.t; /D. / B.t/, where B.t/ is the ball B.0; R.t// in H. By (12.3), the nonautonomous set B D fB.t/gt2R belongs to D and is D -absorbing for the process U.t; /. The uniqueness follows immediately from the fact that A 2 D and from the attracting property. Let us assume that the forcing f is time-independent, with f 2 V 0 . Then, from Theorem 12.1 there follows the existence of the global attractor A H for the semigroup fS.t/gt0 given by S.t/ D U. C t; /, associated with the related autonomous Navier–Stokes problem (cf. Chap. 7). Theorem 12.2. There exists a unique global attractor for the semigroup fS.t/gt0 in H associated with the autonomous Navier–Stokes problem, i.e., problem (12.1) with f 2 V 0 . Actually, the family B.H/ of all bounded sets in H constitutes, for this case, the attraction universe of non-autonomous (and, in this case, constant in time) sets D from Definition 12.2. From (12.4) we conclude that the ball B.0; R/ in H with R D 2=. /kf kV 0 absorbs all bounded sets in H, i.e., S.t/B B.0; R/ for any bounded set B 2 B.H/ and all t t0 .B/. The semigroup fS.t/gt0 is B.H/asymptotically compact, that is, for any sequence tn ! 1 and any sequence xn bounded in H the set fS.tn /xn gn2N is relatively compact in H. Moreover, A D !.B.0; R//—the !-limit set of the ball B.0; R/.

12.2 Construction of the Family of Probability Measures

281

12.2 Construction of the Family of Probability Measures Our main result of this section (Theorem 12.3) concerns the construction of a family f t gt2R of time averaged probability measures in the phase space H and their relation to the pullback attractor fA.t/gt2R . Then (Lemma 12.3) we show how any two measures of the family are related to each other. In Sects. 12.2–12.4 we work with somewhat stronger assumptions on the domain of the flow ˝ and on the forcing f . Namely, we assume that ˝ is an open bounded 2 set in R2 of class C2 and that f 2 Lloc .RI H/, with Z

t

1

e jf ./j2 d < 1

(12.5)

for all t 2 R. 2 Lemma 12.1. Let ˝ be an open bounded set in R2 of class C2 , f 2 Lloc .RI H/ satisfy (12.5), and let fU.t; /g t be the evolutionary process associated with problem (12.1). Then, for every t 2 R there exists a compact set B.t/ absorbing for fU.t; /g t .

Proof. The proof is based on the uniform Gronwall lemma (see Chap. 3) applied to the second energy inequality (see Chap. 7 and also Chap. III, Sect. 2 in [220]) d 2 c kuk2 C kuk2 jf j2 C 3 juj2 kuk4 : dt As usual, we use the first energy inequality (see Chap. 7) d 2 1 juj C kuk2 jf j2 dt

(12.6)

and its consequence ju.t/j2 e.t / ju. /j2 C

et

Z

t 1

e jf ./j2 d;

(12.7)

together with (12.5) to obtain a bound on the norm ku.t/k. However, the bound on the forcing (12.5) is not uniform in t 2 R, and this implies that our estimates are local in time. More precisely, we shall prove that for any u. / 2 D. / for D. / in any non-autonomous set D D fD.t/gt2R 2 D, any fixed t 2 R, and T, r, with 0 < r < T, there exists M.t; T; r/ such that kU.; /u. /k M.t; T; r/

for all

2 Œt T C r; t and

0 .t; T; D/: (12.8)

Since ˝ is a bounded set, V is compactly embedded in H, and (12.8) implies that the closed ball B.t/ D BV .0; M.t; T; r// in V is a compact absorbing set for U.t; /.

282


First, from (12.7) and (12.5) it follows that for all 2 .t T; t/, Z eT t e jf ./j2 d M1 .t; T/ for all 0 .t; T; D/: jU.; /u. /j2 2 1 (12.9) 2 Since f 2 Lloc .RI H/, we have for all s 2 .t T; t r/ and some M2 .t; T; r/ < 1,

1

Z

sCr

jf ./j2 d M2 .t; T; r/:

(12.10)

s

Now, from (12.6) together with (12.9) and (12.10) we have for all s 2 .t T; t r/ and some M3 .t; T; r/ < 1, Z

sCr

ku./k2 d M3 .t; T; r/:

(12.11)

s

In the end, from (12.9) and (12.11) we obtain for all s 2 .t T; t r/ and some M4 .t; T; r/ < 1, c 3

Z

sCr

ju./j2 ku./k2 d M4 .t; T; r/:

s

The last three estimates and the uniform Gronwall lemma give (12.8).

t u

Below we recall the notion of a Banach generalized limit introduced in Chap. 8. Since the existence of a generalized limit follows from the Hahn–Banach theorem which does not guarantee uniqueness, in the sequel we assume implicitly that we work with any given generalized limit. Definition 12.4. By a Banach generalized limit we name any linear functional, denoted LIMT!1 , defined on the space of all bounded real-valued functions on Œ0; 1/ and satisfying (i) LIMT!1 g.T/ 0 for nonnegative functions g. (ii) LIMT!1 g.T/ D limT!1 g.T/ if the usual limit limT!1 g.T/ exists. Theorem 12.3. Let ˝ be an open bounded set in R2 of class C2 and f 2 2 Lloc .RI H/ satisfy (12.5). Let fU.t; /gt be the evolutionary process associated with the Navier–Stokes system (12.1) and fA.t/gt2R be the pullback attractor from Theorem 12.1. Let u0 be an arbitrary element of the phase space H. Then there exists a family of Borel probability measures f t gt2R in the phase space H such that the support of measure t is contained in A.t/ and 1 LIM !1 t

Z

t

Z '.U.t; s/u0 /ds D

'.u/d t .u/ A.t/

for all real-valued and continuous on H functionals ', (' 2 C.H/).

(12.12)

12.2 Construction of the Family of Probability Measures

283

Proof. First we prove a lemma. Lemma 12.2. For u0 2 H and t 2 R the function s ! '.U.t; s/u0 /

(12.13)

is continuous and bounded on .1; t. Proof. We first prove that for any > 0 there exists a ı > 0 such that if jr sj < ı then jU.t; r/u0 U.t; s/u0 j < . We may assume that r < s. It is easy to check (cf. Chap. 7) that if u and v are two solutions of the Navier– Stokes equations in a domain satisfying the Poincaré inequality then Z t 2 2 2 ju.t/ v.t/j ju.0/ v.0/j exp C ku.s/k ds : 0

We have thus, jU.t; r/u0 U.t; s/u0 j2 D jU.t; s/U.s; r/u0 U.t; s/u0 j2 Z t 2 2 jU.s; r/u0 u0 j exp C kU.; s/u0 k d : s

From the continuity of the function r ! U.; r/u0 with values in H it follows that if jr sj < ı is small enough then the right-hand side of the above inequality is as small as needed. Since ' 2 C.H/ we conclude that the function in (12.13) is continuous. As to its boundedness we know that there exists a compact absorbing set B1 .t/ for fU.t; /g, in particular, there exists 0 such that for s 0 , U.t; s/u0 2 B1 .t/. Hence, the function s ! '.U.t; s/u0 / is bounded on .1; 0 . It is also bounded on the compact interval Œ0 ; t. t u We continue the proof of the theorem. Since the function in (12.13) is continuous and bounded, the left-hand side of (12.12) is well defined. We have Z t Z 0 1 1 LIM !1 '.U.t; s/u0 /ds D LIM !1 '.U.t; s/u0 /ds: t t From this equality it follows that the left-hand side of (12.12) depends only on the values of ' on the compact set B1 .t/. By the Tietze extension theorem, the lefthand side of (12.12) defines a linear positive functional on C.B1 .t// and from the Kakutani–Riesz representation theorem it follows that there exists a measure 1t on B1 .t/ defined on some -algebra M in B1 .t/ which contains all Borel sets in B1 .t/ and such that Z Z t 1 '.U.t; s/u0 /ds D '.u/d 1t .u/: LIM !1 t B1 .t/ We can extend this measure easily to all Borel sets in H without enlarging its support.

284


We shall show that the support of the measure 1t is contained in A.t/. Let Bn .t/, n D 1; 2; : : :, be a sequence of compact absorbing sets such that Bn .t/ BnC1 .t/ and \1 nD1 Bn .t/ D A.t/. From the above construction, for every n there exists a measure nt of support contained in Bn .t/ and such that 1 LIM !1 t

Z

t

Z '.U.t; s/u0 /ds D

Bn .t/

'.u/d nt .u/

holds. As the left-hand side of this equation is independent of n, it follows that for ' 2 C.H/, Z B1 .t/

'.u/d 1t .u/ D

Z Bn .t/

'.u/d nt .u/:

(12.14)

By the Tietze extension theorem, every 2 C.Bn .t// is of the form 'jBn .t/ for some ' 2 C.B1 .t//. For any 2 C0 .B1 .t/ n Bn .t// both ' and ' C are continuous extensions of . It then follows that Z

.u/d 1t .u/ D 0:

B1 .t/nBn .t/

For any compact K in B1 .t/ n Bn .t/ there exist 2 C0 .B1 .t/ n Bn .t// such that K on B1 .t/ n Bn .t/ where K is the characteristic function of the set K. We have Z Z 1 0D .u/d t .u/ K .u/d 1t .u/ D 1t .K/ 0; B1 .t/nBn .t/

B1 .t/nBn .t/

whence 1t .K/ D 0 for every compact set K in B1 .t/ n Bn .t/. From the regularity of 1t it follows that 1t .B1 .t/ n Bn .t// D 0. Therefore, the support of 1t is contained in Bn .t/. From (12.14) and the Tietze extension theorem we conclude that Z Bn .t/

'.u/d 1t .u/

Z D Bn .t/

'.u/d nt .u/

for all continuous functions ' on Bn .t/. By the regularity of measures 1t and nt , the above equation holds for all 1t - and nt -integrable functions. Thus 1t D nt on Bn .t/ and on the whole phase space H. In the end, the support of 1t is contained in every set Bn .t/ with n 1 which implies that it is contained in \1 nD1 Bn .t/ D A.t/. The resulting measure with support in A.t/ is denoted by t . Setting ' D 1 in (12.12) we obtain t .A.t// D 1, and so t is a probability measure. The proof of the theorem is complete. t u

12.3 Liouville and Energy Equations

285

Lemma 12.3. The measures f t gt2R satisfy the following relations: t .E/ D .U.t; /1 E/

for all

t; 2 R such that

t

(12.15)

for every Borel set E in the phase space H. Proof. We have for t , and for all functionals ' 2 C.H/, Z

1 '.u/d t .u/ D LIMM!1 tM H D LIMM!1

1 tM

Z

M Z

CLIMM!1 D LIMM!1 Z D

1 M

t

'.U.t; s/u0 /ds '.U.t; s/u0 /ds

M

Z t 1 '.U.t; s/u0 /ds tM Z '.U.t; /U.; s/u0 /ds M

'.U.t; /u/d .u/; H

whence Z

Z '.u/d t .u/ D H

'.U.t; /u/d .u/: H

Using the regularity of t , the Urysohn lemma, and a suitable approximation argument we can extend this relation to all t -integrable functions ' in H and then it becomes equivalent to the generalized invariance property (12.15). t u

12.3 Liouville and Energy Equations In this section we prove that the family of probability measures constructed in Sect. 12.2 satisfies the Liouville equation for a suitable family of generalized moments and also the energy equation. Definition 12.5 (Cf. [96, 99]). By T we denote the class of cylindrical test functionals consisting of the real-valued functionals ˚ D ˚.u/ depending only on a finite number k of components of u, namely, ˚.u/ D ..u; g1 /; : : :; .u; gk //; where is a compactly supported C1 scalar function on Rk and where g1 ; : : :; gk belong to V.

286


Let ˚ 0 denote the differential of ˚ in H, given as ˚ 0 .u/ D

k X

@j ..u; g1 /; : : :; .u; gk //gj ;

(12.16)

jD1

where @j , j D 1; : : :; k, are partial derivatives of on Rk . From (12.16) it follows that ˚ 0 .u/ 2 V. Theorem 12.4. The family of measures f t gt2R satisfies the following Liouville equation: Z

Z ˚.u/d t .u/ A.t/

D

A. /

˚.u/d .u/

Z tZ

.F.u; s/; ˚ 0 .u//d s .u/ds;

(12.17)

A.s/

for all t , where F.u; s/ D f .s/ Au B.u/, and ˚ 2 T . Proof. Let u.t/ be a solution of the Navier–Stokes system du C Au C B.u/ D f .t/ and dt

u.t/jtDs D u0

where

s 2 R:

Since u 2 L2 .; tI D.A// for s < < t, it is not difficult to check that the function t ! .f .t/ Au.t/ B.u.t//; ˚ 0 .u.t/// is integrable on .; t/. From the Navier– Stokes equation it follows that ut 2 L1 .; t; H/. Thus for ˚ 2 T we have d ˚.u.t// D .ut .t/; ˚ 0 .u.t/// dt D .f .t/ Au B.u/; ˚ 0 .u.t/// D .F.u.t/; t/; ˚ 0 .u.t/// almost everywhere in the considered interval. Integrating in t we obtain Z ˚.u.t// ˚.u. // D

t

.F.u./; /; ˚ 0 .u.///d:

For an arbitrary s < , let u0 2 H and u./ D U.; s/u0 for s. Then, for the above solution we have Z t .F.U.; s/u0 ; /; ˚ 0 .U.; s/u0 //d: ˚.U.t; s/u0 / ˚.U.; s/u0 / D

12.3 Liouville and Energy Equations

287

Applying the operator LIMM!1

1 M

Z

.: : :/ds

M

for < we obtain LIMM!1

1 M

Z

˚.U.t; s/u0 /ds LIMM!1

M

Z

1 D LIMM!1 M

Z

t

M

1 M

Z

˚.U.; s/u0 /ds

M

.F.U.; s/u0 ; /; ˚ 0 .U.; s/u0 //dds:

(12.18)

Now, in view of (12.15), (cf. also Chap. II in [228]) the left-hand side equals LIMM!1 Z D

1 M

Z

Z

˚.U.t; /U.; s/u0 /ds

M

Z

˚.U.; /U.; s/u0 /ds

M

Z

˚.U.t; /u/d .u/

˚.U.; /u/d .u/

Z

D

˚.u/d t .u/

˚.u/d .u/:

Similarly, the right-hand side of (12.18) equals LIMM!1

1 M

Z

M

Z

t

.F.U.; s/u0 ; /; ˚ 0 .U.; s/u0 //dds

Z t Z 1 D LIMM!1 .F.U.; s/u0 ; /; ˚ 0 .U.; s/u0 //ds d M M Z t Z 1 0 LIMM!1 D .F.U.; s/u0 ; /; ˚ .U.; s/u0 //ds d: M M For 2 .; t/ we have LIMM!1

1 M

Z

.F.U.; s/u0 ; /; ˚ 0 .U.; s/u0 //ds

M

Z 1 D LIMM!1 .F.U.; /U.; s/u0 ; /; ˚ 0 .U.; /U.; s/u0 //ds M M Z D .F.U.; /u; /; ˚ 0 .U.; /u//d .u/ Z

H

.F.u; /; ˚ 0 .u//d .u/;

D H

288


whence Z Z t 1 .F.U.; s/u0 ; /; ˚ 0 .U.; s/u0 //dds LIMM!1 M M Z tZ D .F.u; /; ˚ 0 .u//d .u/ d:

H

We have used the integrability of the function .t; / .M; / 3 .; s/ ! .F.U.; s/u0 ; /; ˚ 0 .U.; s/u0 // for < , the Fubini theorem, and the linearity of the LIMM!1 operator. In the end, since for every t in R the support of the measure t is contained in A.t/, we arrive at (12.17). t u Theorem 12.5. The family of measures f t gt2R satisfies the following energy equation: Z

juj2 d t .u/ C 2 A.t/

D2

Z tZ

Z tZ

kuk2 d s ds

A.s/

.f .s/; u/d s .u/ds C A.s/

Z A. /

juj2 d .u/ (12.19)

for all t; 2 R, t . Proof. The functionals u ! ˚.u/ D juj2 and u ! kuk2 are bounded (even continuous in the topology of H) on every A.t/, t 2 R. Moreover, ˚ 0 .u/ D 2u. From (12.17) with this ˚ and F.u; s/ D f .s/ Au B.u/ we obtain (12.19) by an approximation argument. Let the functionals ˚m 2 T , m D 1; 2; 3; : : :, be such that ˚m .u/ D jPm uj2 on some ball B.0; R/ in H containing all A./ for t. Here, Pm is the usual Galerkin projection operator in H onto the space spanned by the first m eigenmodes of the Stokes operator. The identity (12.19) thus holds for Pm u in the place of u. Letting m ! 1 and using the Lebesgue dominated convergence theorem we obtain (12.19). t u

12.4 Time-Dependent and Stationary Statistical Solutions A simple corollary from the above considerations is, that assuming any measure of the family f t gt2R to be the initial distribution, the family f t gt forms a statistical solutions of the two-dimensional Navier–Stokes system (Theorem 12.6, cf. Chap. 5 in [99]) for this particular initial data. This solution is unique and it reduces to the stationary time-average statistical solution in the case the forcing

12.4 Time-Dependent and Stationary Statistical Solutions

289

does not depend on time. The support of this stationary solution is contained in the corresponding global attractor (Theorem 12.7, cf. Chap. 8 and also Chap. 4 in [99]). Theorem 12.6. Let fU.t; /gt be the evolutionary process associated with the Navier–Stokes system (12.1) and let u0 be an arbitrary element of the phase space H. Denote by Q 0 the time-average initial measure given by the relation LIM !1

Z

1 0

Z

0

'.U.0; s/u0 /ds D

'.u/d Q 0 .u/ H

for all functionals ' 2 C.H/. Then the support of Q 0 is contained in the pullback attractor’s section A.0/ and the kinetic energy of Q 0 is finite Z

juj2 d Q 0 .u/ < C1:

H

Moreover, the family of measures f t gt0 given by t .E/ D Q 0 .U.t; 0/1 E/ for all Borel sets in H, is the unique statistical solution of the two-dimensional Navier–Stokes system (12.1) with initial distribution Q 0 , namely, it satisfies the following conditions: (S1)

the Liouville equation Z

Z ˚.u/d t .u/ D A.t/

˚.u/d Q 0 .u/ C A.0/

Z tZ 0

.F.u; s/; ˚ 0 .u//d s .u/ds; A.s/

holds for all t 0, F.u; s/ D f .s/ Au B.u/, and the family of moments ˚ 2T, (S2) the energy equation Z

2

juj d t .u/ C 2 A.t/

Z

Z tZ 0

kuk2 d s .u/ds A.s/

2

D

juj d Q 0 .u/ C 2 A.0/

Z tZ 0

holds for all t 0, (S3) the function Z t!

˚.u/d t .u/ A.t/

.f .s/; u/d s .u/ds A.s/

290


is measurable on RC for every bounded and continuous real-valued functional ˚ on H. The function Z t!

juj2 d t .u/

A.t/

is in L1 .RC /. The function Z t!

kuk2 d t .u/

A.t/ 1 .RC /. is in Lloc

If the system (12.1) is autonomous, that is, the forcing f does not depend on time then Theorem 12.6 reduces to the following one. Theorem 12.7. Let fS.t/gt0 be the semigroup associated with the autonomous Navier–Stokes system du C Au C B.u/ D f 2 H dt

with

u.t/jtD0 D u0 ;

and let u0 be an arbitrary element of the phase space H. Then the time-average measure given by the relation 1 LIM !1 0

Z

0

Z 1 '.S.0 s/u0 /ds D LIM !C1 '.S.s/u0 /ds 0 Z D '.u/d .u/ H

for all functionals ' 2 C.H/ has support contained in the global attractor A . Moreover, it is a stationary statistical solution of the considered Navier–Stokes system, namely, the following conditions hold: (ST1) Z

kuk2 d .u/ < 1; H

(ST2) Z A

.F.u; s/; ˚ 0 .u//d .u/ D 0;

where F.u; s/ D f .s/ Au B.u/, and ˚ belongs to T,

12.5 The Case of an Unbounded Domain

291

(ST3) Z E1 juj2
2 kuk .f ; u/ d 0

for all E1 ; E2 such that 0 E1 < E2 1.

12.5 The Case of an Unbounded Domain In this section we do not assume that the domain of the flow is bounded. In consequence, we do not assume the existence of compact absorbing sets for the process U.t; /, the assumption we needed in Sect. 12.2 to construct a family of invariant measures. Moreover, we show that we can obtain time-average invariant measures for U.t; / using as an initial condition any continuous mapping that is contained in the domain of attraction of the pullback attractor. Thus we obtain a twofold generalization of Theorem 12.3. We need the following notion of continuity of the process U.t; /. Definition 12.6. A process U.; / is said to be -continuous if for every u0 in X and every t 2 R, the X-valued function 7! U.t; /u0

(12.20)

is continuous and bounded on .1; t. Our results for the Navier–Stokes equations in certain unbounded domains are direct consequences of the following abstract theorem: Theorem 12.8. Let U.; / be a -continuous evolutionary process in a complete metric space X that has a pullback D-attractor A D fA.t/gt2R . Fix a generalized Banach limit LIMT!1 and let u W R ! X be a continuous map such that u./ 2 D. Then there exists a unique family of Borel probability measures f t gt2R in X such that the support of the measure t is contained in A.t/ and LIM !1

1 t

Z

t

Z '.U.t; s/u.s// ds D

'.v/ d t .v/

(12.21)

A.t/

for any real-valued continuous functional ' on X. In addition, t is invariant in the sense that Z Z '.v/ d t .v/ D '.U.t; /v/ d .v/ for t : (12.22) A.t/

A. /

Remark 12.1. Theorem 12.8 holds for any pullback attractor A D fA.t/gt2R . Assume that there exists a minimal pullback attractor Am D fAm .t/gt2R for the

292


process U.; /. Then the support of measure t from Theorem 12.8 is contained in Am .t/ and in formulas (12.21), (12.22) one can replace A.t/ by Am .t/. In the proof of the theorem we will use the following lemma, cf. [197]: Lemma 12.4. Let .X; / be a metric space, f W X ! R a continuous mapping, and let K X be compact. Then for every > 0 there exists a ı D ıf . / > 0 such that if u 2 X, v 2 K, and .u; v/ ı then jf .u/ f .v/j . Proof (of Theorem 12.8). Fix u./ 2 D and ' 2 C.X/. In view of the existence of the pullback attractor,R the -continuity of the process U.t; / and Lemma 12.4, t 1 the function 7! t '.U.t; s/u.s// ds is bounded on the interval .1; t. It is continuous by an application of the dominated convergence theorem and the Lebesgue differential theorem. It suffices to prove that the function s 7! '.U.t; s/u.s// is bounded on .1; t: it is continuous, since U.; / is -continuous and u./ is continuous, and so it is bounded on every compact interval Œt0 ; t. We now show that it is bounded on .1; t0 for t0 sufficiently large and negative. If this is not the case then there is a sequence fsn g with sn ! 1 such that j'.U.t; sn /u.sn //j ! 1:

(12.23)

However, from the existence of the pullback D-attractor it follows that the process U.; / is pullback D-asymptotically compact, so there exists a subsequence of fsn g (which we relabel) such that U.t; sk /u.sk / ! ! for some ! 2 X (since u./ 2 D). From the continuity of ', we have '.U.t; sk /u.sk // ! '.!/, contradicting (12.23). We now define Z t 1 '.U.t; s/u.s// ds; (12.24) L.'/ D LIM !1 t and claim that L.'/ depends only on the values of ' on A.t/. Indeed, take functions '; 2 C.X/ with ' D on A.t/, and given > 0 use Lemma 12.4 to find a ı such that .u; v/ ı

)

j'.u/ '.v/j C j .u/

.v/j

for every

Now let 0 be such that distX .U.t; s/u.s/; A.t// ı

for all

Then for every s 0 there exists a vs 2 A.t/ such that .U.t; s/u.s/; vs / ı;

s 0 :

v 2 A.t/: (12.25)

12.5 The Case of an Unbounded Domain

293

and so j'.U.t; s/u.s//

.U.t; s/u.s//j

j'.U.t; s/u.s// '.vs /j C j'.vs / C j .vs / using (12.25) and the fact that '.vs / D jL.'

.vs /j

.U.t; s/u.s//j ;

.vs /. Thus

ˇ ˇ Z t ˇ ˇ 1 /j D ˇˇLIM !1 .'.U.t; s/u.s// .U.t; s/u.s/// dsˇˇ t ˇ ˇ Z 0 Z t ˇ ˇ 1 ˇ D ˇLIM !1 .'.U.t; s/u.s// .U.t; s/u.s/// dsˇˇ C t 0 lim sup !1

.0 / t

C lim sup !1

.t 0 / sup fj'.U.t; s/u.s//j C j .U.t; s/u.s//jg : t s2Œ0 ;t

Since > 0 was arbitrary we conclude that L.'/ depends only on the values of ' on A.t/. Define G.'/ D L.l.'// for ' 2 C.A.t//, where l.'/ is an extension of ' given by the Tietze theorem. As G is a linear and positive functional on C.A.t//, we have, by the Kakutani–Riesz representation theorem, Z '.v/ d t .v/:

G.'/ D A.t/

We extend t (by zero) to a Borel measure on X, which we denote again by t , whence for every ' 2 C.X/, Z

Z '.v/ d t .v/ D

'.v/ d t .v/;

A.t/

X

and thus (12.21) holds. To prove (12.22), observe that for t , and for all functionals ' 2 C.X/, Z '.v/ d t .v/ D LIMM!1 X

D LIMM!1

1 tM 1 tM

Z

t

M Z

'.U.t; s/u.s// ds '.U.t; s/u.s// ds

M

C LIMM!1

1 tM

Z

t

'.U.t; s/u.s// ds

294


1 D LIMM!1 M D LIMM!1 Z D

1 M

Z

M Z

'.U.t; /U.; s/u.s// ds Œ' ı U.t; /fU.; s/u.s/g ds

M

Z

Œ' ı U.t; /.v/ d .v/ D

'.U.t; /v/ d .v/;

X

X

where we have used the fact that ' ı U.t; / 2 C.X/.

t u

Now, we can come back to the Navier–Stokes equations. We define the evolutionary process associated to (12.1) and the attraction universe D exactly as in Sect. 12.1. We have already proved the existence of a unique pullback D -attractor A D fA.t/gt2R belonging to D for the process U.t; / in H (cf. Theorem 12.1). By Lemma 12.2 the process U.t; / is -continuous. Thus, the existence of a family of invariant measures t for U.t; / follows from Theorem 12.8. Namely, we have the following result: Theorem 12.9. Let ˝ R2 be an open, bounded or unbounded, set such that there exists a constant 1 > 0 for which Z 1

2

˝

Z

dx

jrj2 dx ˝

for all 2 H01 .˝/:

2 Suppose that f 2 Lloc .RI V 0 / is such that

Z

t

1

" kf ./k2V 0 d < C1

for all

t 2 R:

Let fU.t; /gt be the evolutionary process associated with the Navier–Stokes system (12.1). Fix a generalized Banach limit LIMT!1 and let u W R ! X be a continuous map such that u./ 2 D . Then there exists a unique family of Borel probability measures f t gt2R in the phase space H such that the support of measure t is contained in A.t/ and 1 LIM !1 t

Z

t

Z '.U.t; s/u.s//ds D

'.u/d t .u/ A.t/

for any real-valued continuous functional ' on H. In addition, t is invariant in the sense that Z Z '.v/ d t .v/ D '.U.t; /v/ d .v/ for t : A.t/

A. /


295

12.6 Comments and Bibliographical Notes Theorem 12.1 was proved in [45]. Theorem 12.2 was proved first in [201], cf. also [220]. Theorem 12.8 generalizes Theorem 3.3 from [160] (for the nonautonomous two-dimensional Navier–Stokes equations, using compactness property of those equations) and Theorem 5 from [165] (for autonomous problems, but under weaker assumptions on the dynamical system). Lemma 12.4 comes from [197] stated there as Exercise 10.1, see also Lemma 2 in [165] and Lemma 3.1 in [56]. Theorem 12.9 was proved in [163].

13

Pullback Attractors and Shear Flows

In this chapter we consider the problem of existence and finite dimensionality of the pullback attractor for a class of two-dimensional turbulent boundary driven flows which naturally appear in lubrication theory. We generalize here the results from Chap. 9 to the non-autonomous problem. The new element in our study with respect to that in Chap. 9 is the allowance of the speed of rotation of the cylinder to depend on time. Our aim is first to prove the existence of the unique global in time solution of the problem, and then to study its time asymptotics in the frame of the dynamical systems theory. We use the theory of pullback attractors that allows us to pose quite general assumptions on the velocity of the boundary. Neither quasi-periodicity nor even boundedness is demanded to prove the existence and to estimate the fractal dimension of the corresponding pullback attractor. We shall apply the results from Chap. 11, reformulated here in the language of evolutionary processes.

13.1 Preliminaries In this section we recall a theorem about the existence of pullback attractors for evolutionary processes. Let us consider an evolutionary process U on a metric space X, i.e., a family fU.t; /g1< t
for all

r t:

By P.X/ we denote the family of all nonempty subsets of X. Let D be an attraction universe, i.e., a nonempty class of non-autonomous sets D D fD.t/gt2R such that D.t/ 2 P.X/ for every t 2 R. We have the following result, cf. Theorems 11.4 and 11.5 in Chap. 11. © Springer International Publishing Switzerland 2016 G. Łukaszewicz, P. Kalita, Navier–Stokes Equations, Advances in Mechanics and Mathematics 34, DOI 10.1007/978-3-319-27760-8_13

297

298

13 Pullback Attractors and Shear Flows

Theorem 13.1. Suppose that the process U.t; / is pullback D-asymptotically compact, and that there exists a non-autonomous pullback D-absorbing set for U.; / belonging to the universe D. Then there exists a minimal pullback Dattractor A D fA.t/gt2R for U.; /. Remark 13.1. Observe that there exists at most one pullback D-attractor A 2 D. In fact, if A1 and A2 are two global pullback D-attractors belonging to D, then by the invariance property, distX .A1 .t/; A2 .t// D distX .U.t; /A1 . /; A2 .t//: But then, as A1 2 D and A2 is a global pullback D-attractor, distX .A1 .t/; A2 .t// D lim distX .U.t; /A1 . /; A2 .t// D 0; !1

and therefore A1 .t/ A2 .t/ D A2 .t/: Analogously, one obtains A2 .t/ A1 .t/. Remark 13.2. It is said that the attraction universe D is inclusion-closed if, given the non-autonomous set D 2 D, for any D0 D fD0 .t/gt2R with D0 .t/ 2 P.X/ for t 2 R, such that D0 .t/ D.t/ for all t, one also has D0 2 D: Taking into account that .B; t/ B.t/; and Remark 13.1, we can assert that if the universe D is inclusion-closed and the sets B.t/ are closed, then the pullback D-attractor A constructed in Theorem 13.1 belongs to the universe D, and is the unique pullback D-attractor for U.; / belonging to this universe. The rest of this chapter is organized as follows. In Sect. 13.2 we give a precise formulation of the considered problem and write it in a weak form, suitable for our further considerations. In Sect. 13.3 we derive the first energy inequality and establish the existence of unique global in time solution of the problem. In Sect. 13.4 we prove the existence of a pullback attractor for the corresponding evolutionary process by using the energy equation method. In Sect. 13.5 we obtain an upper bound of the pullback attractor dimension in terms of the data.

13.2 Formulation of the Problem We consider two-dimensional Navier–Stokes equations, ut u C .u r/u C rp D 0;

(13.1)

div u D 0;

(13.2)

13.2 Formulation of the Problem

299

in the channel ˝1 D fx D .x1 ; x2 / W 1 < x1 < 1; 0 < x2 < h.x1 /g; where h is a positive, smooth, and L-periodic function in x1 . Let ˝ D fx D .x1 ; x2 / W 0 < x1 < L; 0 < x2 < h.x1 /g; and @˝ D C [ L [ D , where C and D are the bottom and the top, and L is the lateral part of the boundary of ˝. We are interested in solutions of (13.1) and (13.2) in ˝ which are L-periodic with respect to x1 , or, to be more precise, analogously to Chap. 10, both the velocity u and the Cauchy stress vector Tn on L are L-periodic with respect to x1 . Moreover, we assume uD0

on

D ;

(13.3)

u D U0 .t/e1 D .U0 .t/; 0/

on

C ;

(13.4)

where U0 .t/ is a locally Lipschitz continuous function of time t. Finally, the initial condition at some time 2 R is given as u.x; / D u0 .x/

for

x 2 ˝:

(13.5)

The problem is motivated by a flow in an infinite (rectified) journal bearing ˝ .1; C1/, where D .1; C1/ represents the outer cylinder, and the inner, rotating cylinder is represented by C .1; C1/. We can assume that the rectification does not change the equations as the gap between cylinders is very small with respect to their radii. In our considerations we use the background flow method and transform the boundary condition (13.4) by defining a smooth background flow, a simple version of the Hopf construction, described in detail in Sect. 13.3. Let u.x1 ; x2 ; t/ D U.x2 ; t/e1 C v.x1 ; x2 ; t/; with U.0; t/ D U0 .t/ and

U.h.x1 /; t/ D 0 for

x1 2 .0; L/

and

t 2 R: (13.6)

300


Then v is L-periodic in x1 and satisfies vt v C .v r/v C rp D G on div v D 0

(13.7)

˝ .; 1/;

(13.8)

.D [ C / .; 1/;

(13.9)

on

v D 0 on

˝ .; 1/;

where G D U;x2 x2 e1 U;t e1 Uv;x1 .v/2 U;x2 e1 , and the initial condition v.x; / D v0 .x/ D u0 .x/ U.x2 ; /e1

in ˝:

(13.10)

By .v/2 we denote the second component of v. Now, we define a weak form of the above problem. To this end we need some notations. Let VQ D fv 2 C1 .˝/2 W v is L-periodic in x1 ; div v D 0; v D 0 on D [ C g; V D closure of VQ in H 1 .˝/2 ; H D closure of VQ in L2 .˝/2 : We define the scalar products Z .u; v/ D

˝

u.x/ v.x/dx;

and Z ..u; v// D .ru; rv/ D

ru.x/W rv.x/ dx ˝

in H and V, respectively, and the norms 1

jvj D .v; v/ 2

and

1

kvk D ..v; v// 2 :

Let b.u; v; w/ D ..u r/v; w/; and a.u; v/ D .ru; rv/: Then the natural weak formulation of the problem (13.7)–(13.10) is as follows.

13.3 Existence and Uniqueness of Global in Time Solutions

301

Problem 13.1. Find the velocity v such that for each T > , v 2 C.Œ; TI H/ \ L2 .; TI V/; and for all 2 V and a.e. t > , d .v.t/; / C a.v.t/; / C b.v.t/; v.t/; / D F.v.t/; /; dt

(13.11)

with v.x; / D v0 .x/

on

˝;

where F.v; / D a.; / b.; v; / b.v; ; / .;t ; /;

(13.12)

and D Ue1 is a suitable background flow. In the next section we shall derive the first energy inequality and establish the existence result for the above problem.

13.3 Existence and Uniqueness of Global in Time Solutions To study the pullback attractor associated with a particular problem we first need to know that the latter has a unique global in time solution. In our particular case we have the following existence theorem. Theorem 13.2. Let U0 be a locally Lipschitz continuous function on the real line. Then there exists a unique weak solution of Problem 13.1 such that for all , T, < < T we have v 2 L2 .; T I H 2 .˝// and for each t > the map v0 7! v.t/ is continuous as a map in H. Proof. We shall start the proof from the derivation of the first energy estimate for the solutions, namely, estimate (13.25) below. Taking D v in (13.11), we obtain 1d 2 jvj C a.v; v/ C b.v; v; v/ D F.v; v/: 2 dt Since v D 0 on D [C , v is L-periodic in x1 , and div v D 0, we have b.v; v; v/ D 0, and 1d 2 jvj C kvk2 D F.v; v/: 2 dt

(13.13)

302


Consider the term F.v; v/ on the right-hand side of (13.13). By (13.12) we have F.v; v/ D a.; v/ b.v; ; v/ .;t ; v/:

(13.14)

kvk2 : 8

(13.15)

We have ja.; v/j kkkvk 2kk2 C

To estimate the second term in (13.14) we need the following lemma. Lemma 13.1. For any t 2 R there exists a smooth extension D .x2 ; t/ D U.x2 ; t/e1 D .U.x2 ; t/; 0/ of the boundary condition for u, such that jb.v; .t/; v/j

kvk2 4

for all

v 2 V:

(13.16)

Proof. Let W Œ0; 1/ ! Œ0; 1 be a smooth function such that .0/ D 1 and

supp Œ0; 1=2;

and

max j0 .s/j

p 8:

Let h0 D min0x1 L h.x1 /, and " D ".t/ D

2 =.4h0 jU0 .t/j/

if if

jU0 .t/j =.8h0 /; jU0 .t/j =.8h0 /:

(13.17)

Thus, 0 < ".t/ 2. Define U.x2 ; t/ D U0 .t/.x2 =.h0 ".t///:

(13.18)

Then U matches the boundary conditions (13.6), is L-periodic in x1 , and div Ue1 D 0. Moreover, for Q" D .0; L/ .0; h0 "=2/, ˇ ˇ ˇvˇ jrvjL2 .Q" / jx2 U.x2 ; t/jL1 .Q" / : jb.v; Ue1 ; v/j D jb.v; v; Ue1 /j ˇˇ ˇˇ x2 L2 .Q" / Now, jx2 U.x2 ; t/jL1 .Q" /

h0 " jU0 j; 2

13.3 Existence and Uniqueness of Global in Time Solutions

303

and ˇ ˇ2 ˇ ˇ Z L Z h.x1 / ˇ Z L Z h.x1 / ˇ ˇvˇ ˇ v.x1 ; x2 / ˇ2 ˇ @v.x1 ; x2 / ˇ2 ˇ ˇ ˇ ˇ dx2 dx1 4 ˇ ˇ dx2 dx1 ˇx ˇ 2 ˇ x ˇ ˇ @x ˇ 2 L .Q" / 2 2 0 0 0 0 t u

by the Hardy inequality. Thus, we obtain (13.16).

We estimate the last term on the right-hand side of (13.14) using the Poincaré inequality 1 jvj p kvk 1

for v 2 V;

(13.19)

p p where we can take 1= 1 D hM = 2 with hM D max0x1 L h.x1 /. We get 1 2 j.;t ; v/j j;t jjvj j;t j p kvk j;t j2 C kvk2 : 1 8 1

(13.20)

In view of estimates (13.15), (13.16), and (13.20), jF.v; v/j 2kk2 C

2 j;t j2 C kvk2 ; 1 2

and by (13.13), 1d 2 2 jvj C kvk2 2kk2 C j;t j2 : 2 dt 2 1

(13.21)

To estimate the right-hand side in terms of the data, we use the following lemma. Lemma 13.2. Let U be as in (13.18). Then for almost all t, Z ˝

Z ˝

jU.x2 ; t/j2 dx1 dx2

jU;x2 .x2 ; t/j2 dx1 dx2

1 Lh0 U02 .t/".t/; 2

(13.22)

4LU02 .t/ 1 ; h0 ".t/

(13.23)

and Z

p j"0 .t/j 2 Lh0 ".t/ 0 : jU;t .x2 ; t/j dx1 dx2 jU0 .t/j C 2jU0 .t/j ".t/ 2 ˝ 2

(13.24)

304


Proof. We have jU.x2 ; t/j jU0 .t/j and supp U Q" , which gives (13.22). Moreover, Z

U2 jU;x2 .x2 ; t/j dx D 2 02 h0 " ˝ 2

D

ˇ Z ˇ ˇ 0 x2 ˇ2 ˇ dx ˇ ˇ h" ˇ

U02 L h20 "2

0

˝

Z

h0 " 2

0

ˇ ˇ 2 ˇ 0 x2 ˇ2 ˇ dx2 4LU0 1 ; ˇ ˇ ˇ h0 " h0 "

p as j0 j 8, which gives (13.23). In order to obtain (13.24) we use (13.18) and the definition of to get ˇ ˇˇ ˇ ˇ x2 x2 1 ˇˇ x2 "0 .t/ ˇˇ ˇˇ 0 ˇ C jU0 .t/j 2 ˇ jU;t .x2 ; t/j ˇ ˇ h0 ".t/ " .t/ h0 h0 ".t/ ˇ p j"0 .t/j jU00 .t/j C 2jU0 .t/j ".t/ jU00 .t/j

for 0 x2 h0 "=2 and almost all t, from which the assertion follows easily.

t u

Applying now Lemma 13.2 to inequality (13.21) with " as in (13.17) we obtain the first energy estimate, 1d jv.t/j2 C kv.t/k2 f 2 .t/ 2 dt 2

a.e. in t;

(13.25)

f 2 .t/ D c1 .; ˝/.1 C jU0 .t/j3 C jU00 .t/j2 /:

(13.26)

where

From (13.25) and (13.19) we conclude further that 1d jv.t/j2 C jv.t/j2 f 2 .t/; 2 dt 2 with D 1 , and that 2

.t /

jv.t/j e

2

jv./j C 2

Z

t

e.st/ f 2 .s/ds:

(13.27)

To prove the uniqueness of solutions and their continuous dependence on the initial data we assume that v and vN are two solutions of Problem 13.1, write (13.11) for their difference, w D v v, N and set D w, to get 1d jw.t/j2 C kw.t/k2 C b.w; v; w/ D b.w; ; w/: 2 dt 2

13.4 Existence of the Pullback Attractor

305

Now we use the Ladyzhenskaya inequality (cf. Lemma 10.2), kvkL4 .˝/ c2 .˝/1=4 jvj1=2 kvk1=2 for all v 2 V (we can take c2 .˝/ D 18 maxf2; 1 C .hM =L/2 g), to get jb.w; v; w/j kwkL4 .˝/ kvkkwkL4 .˝/ c2 .˝/1=2 jwjkwkkjvk

1 2 kwk2 C c2 .˝/jwj2 kvk2 : 8

From this, together with (13.16), we get d 4 jw.t/j2 C kw.t/k2 c2 .˝/kvk2 jwj2 ; dt 4 and d 4 jw.t/j2 C jw.t/j2 c2 .˝/kvk2 jwj2 : dt 4 By (13.25) and (13.27) the function s ! kv.s/k2 is locally integrable on the real line, whence the Gronwall lemma gives Z t 4 (13.28) jw.t/j2 jw. /j2 exp c2 .˝/kv.s/k2 ds : 4 This means that the map v. / ! v.t/ for t > is continuous in H, and, in particular, that the considered problem is uniquely solvable. The existence part of the proof is based on inequality (13.25), the Galerkin approximations, and the compactness method, and is very similar to the proof of Theorem 9.6. t u

13.4 Existence of the Pullback Attractor In this section we prove the existence of the pullback attractor for the evolutionary process associated with the considered problem. For t let us define the map U.t; / in H by U.t; /v0 D v.tI ; v0 /

for t

and

v0 2 H;

(13.29)

where v.tI ; v0 / is the solution of Problem 13.1. From the uniqueness of solution to this problem, one immediately obtains that U.t; /v0 D U.t; r/U.r; /v0 ;

for all

rt

and

v0 2 H:

306


From Theorem 13.2 it follows that for all t ; the mapping U.t; / W H ! H defined by (13.29) is continuous. Consequently, the family fU.t; /g t defined by (13.29) is a process in H. Moreover, it can be proved that U.; t/ is sequentially weakly continuous, more exactly, we have the following result. Proposition 13.1. Let fv0n g H be a sequence converging weakly in H to an element v0 2 H: Then U.t; /v0n ! U.t; /v0 U.; /v0n ! U.; /v0

for all t ;

weakly in H

weakly in L2 .; TI V/

for all

T > :

Proof. The proof is almost identical to that of Lemma 2:1 in [201] and we omit it. t u Now, we define the universe of the non-autonomous sets. For D 1 , let D D fD W R ! P.H/ W

lim et jD.t/j2 D 0g;

t!1

(13.30)

where jD.t/j D supfjyj W y 2 D.t/g. We can prove the following result: Theorem 13.3. Let U0 be a locally Lipschitz continuous function on the real line such that Z t es .jU0 .s/j3 C jU00 .s/j2 / ds < C1 for all t 2 R: (13.31) 1

Then, there exists a unique pullback D -attractor A 2 D for the process U.t; / defined by (13.29). Proof. Let D 2 D be given. From the first energy estimate (13.27) and from (13.31) we obtain 2

.t /

jU.t; /v0 j e

2

t

Z

jD. /j C 2e

t 1

es f .s/2 ds < 1

for all v0 2 D. /, and all t . Denote by R .t/ the positive number given for each t 2 R by .R .t//2 D 3et

Z

t

1

es f .s/2 ds;

(13.32)


307

and consider the non-autonomous set B D fB .t/gt2R where B .t/ are closed balls in H defined by B .t/ D fw 2 H W jwj R .t/g: In view of (13.30) it is immediate to see that B 2 D . Moreover, by (13.30) and (13.32), the family B is pullback D -absorbing for the process U.t; /. Clearly, the attraction universe D is inclusion-closed and the sets B .t/ are closed, and thus, according to Theorem 13.1 and Remark 13.2, to complete the proof of the theorem we have to prove that U.t; / is pullback D -asymptotically compact. Let D D fD.t/gt2R 2 D , a sequence n ! 1, a sequence v0n 2 D.n /, and t 2 R be fixed. We must prove that we can extract a subsequence that converges in H from the sequence fU.t; n /v0n g. As the non-autonomous set B D fB .t/gt2R is pullback D -absorbing, for each integer k 0 there exists a D .k/ t k such that U.t k; /D. / B .t k/

for all

D .k/:

(13.33)

It is not difficult to deduce from (13.33), by a diagonal procedure, the existence of a subsequence f.n0 ; v0n0 /g f.n ; v0n /g; and a sequence fwk g1 kD0 H; such that for all k D 0; 1; 2; : : : we have wk 2 B .t k/ and U.t k; n0 /v0n0 ! wk

weakly in

H:

Now, observe that, by Proposition 13.1, w0 D weak 0lim U.t; n0 /v00n n !1

D weak 0lim U.t; t k/U.t k; n0 /v0n0 n !1

D U.t; t k/.weak 0lim U.t k; n0 /v0n0 /; n !1

i.e., U.t; t k/wk D w0

for all

k D 0; 1; 2; : : : :

(13.34)

Then, by the weak lower semicontinuity of the norm, inf jU.t; n0 /v0n0 j: jw0 j lim 0 n !1

If we now prove that also lim sup jU.t; n0 /v0n0 j jw0 j; n0 !1

(13.35)

308


then we will have lim jU.t; n0 /v0n0 j D jw0 j;

n0 !1

and this, together with the weak convergence, will imply the strong convergence in H of U.t; n0 /v0n0 to w0 . Our aim is to prove (13.35). We shall use the notation, hg.t/; wi D a..t/; w/ .;t ; w/

for w 2 V;

and Q.t; w/ D kwk2

2 jwj C b.w; .t/; w/ 2

for

.t; w/ 2 R V:

Then, the energy equation (13.13) can be written as d 2 jvj C jvj2 D 2.hg.t/; vi Q.t; v//; dt whence for all t ; and all v0 2 H, (13.36) jU.t; /v0 j2 D jv0 j2 e.t/ Z t e.st/ .hg.s/; U.s; /v0 i Q.s; U.s; /v0 // ds; C2

and, thus, for all k D 0; 1; 2; : : : and all n0 t k; jU.t; n0 /v0n0 j2 D jU.t; t k/U.t k; n0 /v0n0 j2

(13.37)

D ek jU.t k; n0 /v0n0 j2 Z t C2 e.st/ hg.s/; U.s; t k/U.t k; n0 /v0n0 i ds Z 2

tk t

e.st/ Q.s; U.s; t k/U.t k; t n0 /v0n0 / ds:

tk

As, by (13.33), U.t k; n0 /v0n0 2 B .t k/

n0 D .k/

and

k D 0; 1; 2; : : : ;

lim sup ek ŒU.t k; n0 /v0n0 2 ek R2 .t k/

for

k D 0; 1; 2; : : : :

for all

we have n0 !1

(13.38)


309

On the other hand, as U.t k; n0 /v0n0 ! wk weakly in H, from Proposition 13.1 we have U.; t k/U.t k; n0 /v0n0 ! U.; t k/wk

(13.39)

weakly in L2 .t k; tI V/: Taking into account that, in particular, e.t/ g./ 2 L2 .t k; tI V 0 /; we obtain from (13.39), Z

t

lim 0

n !1 tk

e.st/ hg.s/; U.s; t k/U.t k; n0 /v0n0 i ds Z

t

D

e.st/ hg.s/; U.s; t k/wk i ds:

(13.40)

tk

Moreover, it is not difficult to see, using (13.16), the continuous injection of H 1 .˝/ into L4 .˝/; and the fact that by (13.23), r 2 L1 .t k; tI L2 .˝/22 /; that the functional Z t e.st/ Q.s; w.s// ds .w/ D tk

is convex and continuous, and consequently weakly lower semicontinuous on the space L2 .t k; tI V/: Thus, we also obtain from (13.39) Z

t

e.st/ Q.s; U.s; t k/wk / ds

tk

Z lim inf 0 n !1

t

(13.41)

e.st/ Q.s; U.s; t k/U.t k; n0 /v0n0 / ds:

tk

Then, from (13.37), (13.38), (13.40), and (13.41), we obtain lim sup jU.t; n0 /v0n0 j2 ek R2 .t k/ n0 !1

Z C2

t

(13.42)

e.st/ .hg.s/; U.s; t k/wk i Q.s; U.s; t k/wk // ds:

tk

Now, from (13.34) and (13.36), jw0 j2 D jU.t; t k/wk j2 D jwk j2 ek (13.43) Z t 2 e.st/ .hg.s/; U.s; t k/wk i Q.s; U.s; t k/wk // ds: tk

310


From (13.42) and (13.43), we have lim sup jU.t; n0 /v0n0 j2 ek R2 .t k/ C jw0 j2 jwk j2 ek n0 !1

ek R2 .t k/ C jw0 j2 ; and thus, taking into account that ek R2 .t k/ D

3et

Z

tk

1

es kg.s/k2V 0 ds ! 0;

when k ! C1, we easily obtain (13.35) from the last inequality.

t u

13.5 Fractal Dimension of the Pullback Attractor In this section we prove Theorem 13.4. Let U0 be a locally Lipschitz continuous function on the real line such that for some t? 2 R, r > 0, Mb > 0, M > 0, all t t? , and all s t? r, Z jU0 .t/j Mb

sCr

and s

jU00 ./j2 d M:

(13.44)

Then the attractor A D fA.t/gt2R from Theorem 13.3 has finite fractal dimension, namely, dfH .A.t// d

(13.45)

for all t 2 R and some constant d. To this end we shall use a result from [42] which in our notation can be expressed as follows. Theorem 13.5. Suppose there exist constants K0 ; K1 , > 0 such that jA.t/j D supfjyj W y 2 A.t/g K0 jtj C K1

(13.46)

for all t 2 R. Assumepthat for any t 2 R there exist T D T.t/, l D l.t; T/ 2 Œ1; C1/, ı D ı.t; T/ 2 .0; 1= 2/, and N D N.t/ such that for any u; v 2 A. / and t T, jU. C T; /u U. C T; /vj lju vj;

(13.47)

jQN .U. C T; /u U. C T; /v/j ıju vj;

(13.48)

13.5 Fractal Dimension of the Pullback Attractor

311

where QN is the projector mapping from H onto some subspacep HN? of codimension p N 2 N. Then, for any D .t/ > 0 such that D .t/ D .6 2l/N . 2 / < 1, the fractal dimension of A.t/ is bounded, with dfH .A.t// N C :

(13.49)

Proof (of Theorem 13.4). First we show that for some M0 > 0 and all t t? , kA.t/k D supfkyk W y 2 A.t/g M0 :

(13.50)

Here we need the second energy estimate. With D Av in (13.11) we get 1d kv.t/k2 C jAv.t/j2 C b.v.t/; v.t/; Av/ D F.v.t/; Av/; 2 dt

(13.51)

where F.v; Av/ D .; Av/ b.; v; Av/ b.v; ; Av/ .;t ; Av/: To estimate the nonlinear term on the left-hand side of (13.51) we use the Agmon inequality kvkL1 .˝/ cjvj1=2 jAvj1=2 ; that holds for v 2 D.A/. This estimate follows from the usual Agmon inequality 1=2

1=2

kwkL1 .˝3 / ckwkL2 .˝3 / kwkH 2 .˝3 / ; where ˝3 is a smooth set such that ˝ ˝3 ˝1 and w D 'v, ' being the suitable cut-off function, and the inequality kvkH 2 .˝/ cjAvj for v 2 D.A/, cf. the proof of Theorem 9.6. We have thus jb.v; v; Av/j kvkL1 .˝/ kvkjAvj cjvj1=2 kvkjAvj3=2

c0 jAvj2 C 3 jvj2 kvk4 : 8

Moreover, j.;t ; Av/j

2 jAvj2 C j;t j2 ; 8

and jb.; v; Av/j kkL1 .˝/ kvkjAvj

2 jAvj2 C kk2L1 .˝/ kvk2 : 8

312


Similarly, 2 jAvj2 C krk2L1 .˝/ jvj2 : 8

jb.v; ; Av/j jvjkrkL1 .˝/ jAvj Finally, j.; Av/j

jAvj2 C 2jj2 : 8

Therefore, from (13.51) we obtain 1d 1 kv.t/k2 C jAv.t/j2 a.t/kv.t/k2 C b.t/; 2 dt 2

(13.52)

where c0 2 jv.t/j2 kv.t/k2 C k.t/k2L1 .˝/ ; 3

a.t/ D

(13.53)

and b.t/ D

2 2 j;t .t/j2 C kr.t/k2L1 .˝/ jv.t/j2 C 2Œ.t/2 :

(13.54)

Now, recall that for all v0 2 A. / and all t , we have jU.t; /v0 j2 e.t / jA. /j2 C 2

Z

t 1

e.st/ f 2 .s/ds:

(13.55)

Let us fix some T > r and an arbitrary t t . Let v0 2 A. / for near 1. Then, from (13.55) and (13.30) we deduce that for all 2 .t T; t/, jv./j2 3eT

Z

t 1

e.st/ f 2 .s/ds

1 D 3e C1 .; ˝/ C et T

(13.56) Z

t 1

s

3

e ŒjU0 .s/j C

jU00 .s/j2 ds

:

From (13.54) we have Z s

sCr

b./d

Z p Z sCr 0 2Lh0 8 sCr .3 C 2 2/ jU0 ./2 d C 3 jU0 ./j4 /jv./j2 d s s L.4h0 /3 C 2

!2 Z

sCr

sup 00 ./ 2R

s

jU0 ./j5 d

if U0 ./

; 8h0


313

and Z

sCr

Z

sCr

b./d Lh0

s

s

L 3 C 3 2 8 h0

jU00 ./j2 d C !2

3 8 h40

00

sup ./

Z

sCr

jv./j2 d

s

if U0 ./

2R

: 8h0

Thus, by (13.44) and (13.56) we conclude that there exists M2 .r/ such that Z

sCr

b./d M2 .r/

s

for all s 2 .t T; t r/. From (13.25), (13.26), (13.31), (13.44), and (13.56) we deduce further that there exists M3 .r/ such that Z

sCr

kv./k2 d M3 .r/

(13.57)

s

for all s 2 .t T; t r/. Finally, by (13.53) together with (13.56), (13.57), and (13.44), there exists M1 .r/ such that Z

sCr

a./d M1 .r/

(13.58)

s

for all s 2 .t T; t r/. With these estimates we apply now the uniform Gronwall lemma to inequality (13.52) to get, in particular, 2

kv.t/k

M3 .r/ Q C 2M2 .r/ exp 2M1 .r/ M.r/; r

which gives (13.50). We shall prove now that in our case of the Navier–Stokes flow inequality (13.46) holds true for t t? , and inequalities (13.47) and (13.48) hold true for C T t t? . This restriction for the time variable t in the hypotheses of Theorem 13.5 produces a similar change in its assertion, namely, inequality (13.49) holds then true for t t? . Such restricted result is, however, sufficient in our case of the Navier– Stokes flow in order to obtain (13.45) for all t as it will be shown in the end of the proof. Observe that in view of (13.50) inequality (13.46) is satisfied for t t? , namely, 1 jA.t/j p M0 1

for

t t? :

314


Inequality (13.47) for C T t t? follows from inequality (13.28). In fact, let v. / and v. N / be in A. /, and denote ! D v v. N Then from (13.28) and (13.50) it follows that j!. C T/j2 l.T/j!. /j2 ;

(13.59)

with l.T/ D expf. 4 C 4 c2 .˝/M02 /Tg (we may always assume that the constant l.T/ in (13.59) is greater or equal than one). Now, we shall prove that inequality (13.48) holds true. Multiplying d! C A! C B.!; v/ N C B.v; !/ D B.w; / dt P by Qm ! defined by Qm !.x; t/ D 1 Q j .x/ (Qm D I Pm , Pm being the jDmC1 !.t/w projection on the space spanned by the first m eigenfuntions of the Stokes operator with the periodic boundary conditions on L and Dirichlet boundary conditions on C [ D ), we obtain 1d N !; Qm !/ jQm !j2 C kQm !k2 C b.!; v; Qm !/ C b.v; 2 dt D b.w; ; Qm !/:

(13.60)

Our first step is to obtain a differential inequality for jQm !j2 of the form (13.65) below. To this end we have to estimate the nonlinear terms in (13.60). We have, b.!; v; Qm !/ D b.Pm !; v; Qm !/ C b.Qm !; v; Qm !/: Now, jb.Pm !; v; Qm !/j D jb.Pm !; Qm !; v/j 1

1

1

1

1

c2 .˝/ 2 jPm !j 2 kPm !k 2 kQm !kjvj 2 kvk 2 ; and kPm !k2 m jPm !j2 , whence, from (13.50) we conclude jb.Pm !; v; Qm !/j

2c2 .˝/M02 12 2 m j!j C kQm !k2 : p 8 1

(13.61)

jb.Qm !; v; Qm !/j

2c2 .˝/M02 jQm !j2 C kQm !k2 : 8

(13.62)

We have also


315

We estimate the last term on the left-hand side of (13.60) as follows, jb.v; N !; Qm !/j D jb.v; N Pm !; Qm !/j 1

1

1

1

1

c2 .˝/ 2 jvj 2 kvk N 2 jPm !j 2 kPm !k 2 kQm !k

2c2 .˝/M02 12 2 m j!j C kQm !k2 : p 8 1

(13.63)

Using (13.44) we treat the term on the right-hand side of (13.60) as follows, jb.!; ; Qm !/j D jb.!; Qm !; /j kkL1 .˝/ j!jkQm !k

2 2 2 M j!j C kQm !k2 : b 8

(13.64)

From (13.60) together with (13.61)–(13.64), and in view of mC1 jQm !j2 kQm !k2 ; we obtain, in particular, for s C T t t? , d jQm !.s/j2 C m jQm !.s/j2 m j!.s/j2 ; ds

(13.65)

where 4c2 .˝/M02 m D mC1

and

m D

4Mb2 8c2 .˝/M02 12 : m C p 1

We can see that for m large enough we have m > 0. Now, using the bound j!.s/j2 l.T/j!. /j2 for s C T, we obtain d jQm !.s/j2 C m jQm !.s/j2 m l.T/j!. /j2 : ds Multiplying both sides by expfm .s /g we get d fjQm !.s/j2 expfm .s /gg m l.T/j!. /j2 expfm .s /g: ds Integrating with respect to s in the interval .; C T/ we obtain, jQm !. C T/j2 ı 2 .T/j!. /j2 ; with ı 2 D expfm Tg C

m l.T/ 1 < m 2

316


for m large enough, as m is of the order of mC1 . This gives inequality (13.48) and thus completes the proof of a finite dimensionality of the sets A.t/, for t t? , with dfH .A.t// m C : The same estimate holds also for t > t? in view of the property of the fractal dimension under Lipschitz continuous mappings, cf. Theorem 9.1. t u

13.6 Comments and Bibliographical Notes To prove the existence of the pullback attractor we used, in Sect. 13.4, the energy equation method, developed in [44, 45] for the pullback attractor theory. This method works also in the case of certain unbounded domains of the flow as it bypasses the usual compactness argument. In turn, to estimate the pullback attractor dimension we used, in Sect. 13.5, the method proposed in [42], alternative to the usual one [149, 220]. It is also possible to study the finite dimensionality of pullback attractors through so-called pullback exponential attractors, see [51, 52, 151]. When h D const and U0 D const, problem (13.1)–(13.5) was intensively studied in several contexts. Most results concern the autonomous Navier–Stokes equations (cf. Sect. 9.4). In [89], the domain of the flow is an elongated rectangle ˝ D .0; L/ .0; h/, L h. In this case the attractor dimension can be estimated from above by c Lh Re3=2 , where c is a universal constant and Re D Uh is the Reynolds number. In [241] optimal bounds of the attractor dimension are given for a flow in a rectangle .0; 2L/ .0; 2L=˛/, with periodic boundary conditions and given external forcing. The estimates are of the form c0 =˛ dfH .A / c1 =˛, see also [180]. Some free boundary conditions are considered in [242], see also [222], and an upper bound on the attractor dimension established with the use of a suitable anisotropic version of the Lieb–Thirring inequality, similarly to [89]. Dirichletperiodic and free-periodic boundary conditions and domains with more general geometry were considered in [22, 24] (cf. also [25]) where still other forms of the Lieb–Thirring inequality were established to study the dependence of the attractor dimension on the shape of the domain of the flow. Slip boundary condition and unbounded domain case were considered in [183]. The autonomous case with h ¤ const was considered in [22, 24]. Boundary driven flows in smooth and bounded two-dimensional domains for a non-autonomous Navier–Stokes system were considered in [178], by using an approach of Chepyzhov and Vishik, cf. [62]. An extension to some unbounded domains can be found in [181], cf. also [164].

14

Trajectory Attractors and Feedback Boundary Control in Contact Problems

I hail a semigroup when I see one and I seem to see them everywhere. Friends have observed, however, that there are mathematical objects which are not semigroups. – Einar Hille

In this chapter we consider two-dimensional nonstationary incompressible Navier–Stokes shear flows with nonmonotone and multivalued leak boundary conditions on a part of the boundary of the flow domain. Our considerations are motivated by feedback control problems for fluid flows in domains with semipermeable walls and membranes and by the theory of lubrication. Our aim is to prove the existence of global in time solutions of the considered problem which is governed by a partial differential inclusion, and then to prove the existence of a trajectory attractor and a weak global attractor for the associated multivalued semiflow.

14.1 Setting of the Problem The problem we consider is as follows. The flow of an incompressible fluid in a two-dimensional domain ˝ is described by the equation of motion ut u C .u r/u C rp D 0 for

.x; t/ 2 ˝ RC ;

(14.1)

where the viscosity coefficient > 0, and the incompressibility condition div u D 0 for

.x; t/ 2 ˝ RC :

(14.2)

To define the domain ˝ of the flow let us consider the channel ˝1 D fx D .x1 ; x2 / W 1 < x1 < 1; 0 < x2 < h.x1 /g;


317

318

14 Trajectory Attractors and Feedback Boundary Control in Contact Problems

where the function h W R ! R is positive, smooth, and L-periodic. Then we set ˝ D fx D .x1 ; x2 / W 0 < x1 < L; 0 < x2 < h.x1 /g; and @˝ D C [ L [ D , where C and D are the bottom and the top, and L is the lateral part of the boundary of ˝. The domain ˝ is schematically presented in Fig. 14.1. By n we will always denote the unit normal vector on @˝. Fig. 14.1 Schematic overview of the domain ˝ and parts C ; D ; L of its boundary

We are interested in solutions of (14.1)–(14.2) such that the velocity u and the Cauchy stress vector Tn, where T is the stress tensor given by the constitutive law Tij D pıij C .ui;j C uj;i /, are L-periodic with respect to x1 on L . We assume that uD0

on

D RC :

(14.3)

On the bottom C we impose the following conditions. The tangential component u of the velocity vector on C is given, namely, for some s 2 R, we have u D u un n D .s; 0/

on

C RC ;

where

un D u n:

(14.4)

Furthermore, we assume the following subdifferential boundary condition pQ .x; t/ 2 @j.un .x; t//

on

C RC ;

(14.5)

where pQ D p C 12 juj2 is the total pressure (called the Bernoulli pressure), j W R ! R is a given locally Lipschitz potential, and @j is a Clarke subdifferential of j (see Sect. 3.7 and [72, 84] for the definition and properties of Clarke subdifferential). Let, moreover, u.0/ D u0

in ˝:

(14.6)

The considered problem is motivated by the examination of a certain twodimensional flow in an infinite (rectified) journal bearing ˝.1; C1/, where the

14.2 Weak Formulation of the Problem

319

outer cylinder is represented by D .1; C1/, and C .1; C1/ represents the inner, rotating cylinder. In the lubrication problems the gap h between cylinders is never constant. We can assume that the rectification does not change the equations as the gap between cylinders is very small with respect to their radii. A physical interpretation of the boundary condition (14.5) can be as follows. The potential j in our control problem is not convex as its subdifferential corresponds to the nonmonotone relation between the normal velocity un and the total pressure pQ at C . Assuming that, left uncontrolled, the total pressure at C would increase with the increase of the normal velocity of the fluid at C , we control pQ by a hydraulic device which opens wider the boundary orifices at C when un attains a certain value and thus pQ drops at this value of un . Particular examples of such relations are provided in [175, 176].

14.2 Weak Formulation of the Problem In this section we introduce the basic notations and define a notion of a weak solution u of the initial boundary value problem (14.1)–(14.6). For convenience, we shall work with a homogeneous problem whose solution v has the tangential component v at C equal to zero, and then u D v C w for a suitable extension w of the boundary data. In order to define a weak formulation of the homogeneous problem (14.1)–(14.6) we need to introduce some function spaces and operators. Let W D fw 2 C1 .˝/2 W div w D 0 in ˝; w is L-periodic in x1 ; w D 0 on D ; w D 0 on C g; and let V and H be the closures of W in the norms of H 1 .˝/2 and L2 .˝/2 , respectively. In the sequel we will use the notation k k; j j to denote, respectively, the norms in V and H. We denote the trace operator V ! L2 .C /2 by . By the trace theorem, is linear and bounded; we will denote its norm by k k WD k kL .VIL2 .C /2 / . In the sequel we will write u instead of u for the sake of notation simplicity. Let the operators A W H 1 .˝/2 ! V 0 and BŒ W H 1 .˝/2 ! V 0 be defined by Z hAu; vi D

˝

rot u rot v dx

(14.7)

for all u 2 H 1 .˝/2 , v 2 V, and BŒu D B.u; u/, where Z hB.u; w/; zi D ˝

for u; w 2 H 1 .˝/2 , z 2 V.

.rot u w/ z dx

(14.8)

320


According to the hydrodynamical interpretation of the considered problem given in Sect. 14.1, we can understand the rot operator in the following way. For u.x1 ; x2 / D .u1 .x1 ; x2 /; u2 .x1 ; x2 // and uN .Nx/ D .u1 .x1 ; x2 /; u2 .x1 ; x2 /; 0/, where xN D .x1 ; x2 ; x3 /, we set rot u.x1 ; x2 / D rot uN .Nx/. Let G D ˝ .0; 1/ and f .Nx/ D f .x1 ; x2 / be a scalar function. Then Z

Z ˝

f .x1 ; x2 / dx D

f .Nx/ dNx:

(14.9)

G

In particular, for u; v in V, Z

Z hAu; vi D Z

˝

rot u.x/ rot v.x/ dx D Z r uN .Nx/ r v.N N x/ dNx D

D

rot uN .Nx/ rot v.N N x/ dNx G

G

ru.x/ rv.x/ dx:

(14.10)

˝

To work with the boundary condition (14.5) we rewrite equation of motion (14.1) in the Lamb form, ut C rot rot u C rot u u C r pQ D 0 in ˝ RC :

(14.11)

Further, to transform the problem into the homogeneous one, for u 2 H 1 .˝/2 let w 2 H 1 .˝/2 be such that div w D 0, w D 0 at D , w is L-periodic on L , and w D u D s and wn D 0 on C , and let u D v C w. Then v 2 V as v D 0 at C . Moreover, vn D un on C . Multiplying (14.11) by z 2 V and using the Green formula we obtain hv 0 .t/ C Av.t/ C BŒv.t/; zi C

Z C

pQ zn dS D hF; zi C hG.v/; zi;

(14.12)

where Z hF; zi D

rot w rot z dx hB.w; w/; zi;

(14.13)

˝

and hG.v/; zi D hB.v; w/ C B.w; v/; zi: Above we have used the formula Z Z Z rot R a dx D R rot a dx C ˝

˝

@˝

.R a/ n dS;

(14.14)

(14.15)

14.2 Weak Formulation of the Problem

321

with R D rot v or R D rot w. Formula (14.15) is easy to get using the threedimensional vector calculus and (14.9). Observe that if a D 0 on @˝, then we have .R a/ n D .R an n/ n D 0: We need the following assumptions on the potential j: .j1/ j W R ! R is locally Lipschitz, .j2/ @j satisfies the growth condition jj c1 C c2 juj for all u 2 R and all 2 @j.u/, with c1 > 0 and c2 > 0, .j3/ @j satisfies the dissipativity inf[email protected]/ u d1 d2 juj2 , for all

condition u 2 R, where d1 2 R and d2 2 0; kk 2 .

From (14.12) we obtain the following weak formulation of the homogeneous problem. Problem 14.1. Let v0 2 H. Find 2 1 .RC I V/ \ Lloc .RC I H/; v 2 Lloc 4

3 with v 0 2 Lloc .RC I V 0 /, v.0/ D v0 , and such that

hv 0 .t/ C Av.t/ C BŒv.t/; zi C ..t/; zn /L2 .C / D hF; zi C hG.v.t//; zi; (14.16) 2 ; .t/ 2 [email protected] n .;t//

for a.e. t 2 RC and for all z 2 V. In the above definition we use the notation SU2 D fu 2 L2 .C / W u.x/ 2 U.x/

for a.e.

x 2 C g;

2 .RC I V/ valid for a multifunction U W C ! 2R . Note that from the fact that v 2 Lloc 4

3 and v 0 2 Lloc .RC I V 0 / it follows that v 2 C.RC I V 0 /, and hence the initial condition makes sense. We define Cw .Œ0; TI H/ as the space of all functions u W Œ0; T ! H which are weakly continuous, i.e., such that for any 2 H the scalar product 1 .u.t/; / in H is a continuous function of t for t 2 Œ0; T. Since v 2 Lloc .RC I H/ C then (see Lemma 3.18 or [227], Theorem II.1.7) v 2 Cw .R I H/, and thus the initial condition makes sense in the phase space H. 2 One can see (cf. [176]) that if v 2 Lloc .RC I V/ is a sufficiently smooth solution of the partial differential inclusion (14.16) then there exists a distribution pQ such that the conditions (14.11) and (14.5) hold for u D v C w. In conclusion, the function u D v C w can be regarded as a weak solution of the initial boundary value problem (14.1)–(14.6), provided v is a solution of Problem 14.1 with v.0/ D u0 w, u0 2 H.

322


The trajectory space K C of Problem 14.1 is defined as the set of those of its solutions with some v0 2 H that satisfy the following inequality, 1d jv.t/j2 C C1 kv.t/k2 C2 .1 C kFk2V 0 /; 2 dt

(14.17)

where C1 ; C2 > 0 and inequality (14.17) is understood in the sense that for all 0 t1 < t2 and for all 2 C01 ..t1 ; t2 //, 0 we have 1 2

Z

t2 t1

jv.t/j

2

0

Z .t/ dt C C1

t2

2

kv.t/k

.t/ dt C2 .1 C

t1

kFk2V 0 /

Z

t2 t1

.t/ dt: (14.18)

Note that since we cannot guarantee that for every solution of Problem 14.1 hv 0 .t/; v.t/i D 12 dtd jv.t/j2 holds, we cannot derive (14.17) for every solution of Problem 14.1. In the next section we prove that the trajectory space is not empty.

14.3 Existence of Global in Time Solutions In this section we give a proof of the existence of solutions of Problem 14.1 that satisfy inequality (14.17). The proof will be based on the standard technique that uses the regularization of the multivalued term and in main points will follow the existence proofs in Chap. 10 (see also [162, 176]). The operators A and B defined in (14.7) and (14.8) and restricted to V have the following properties: (1)

A W V ! V 0 is a linear, continuous, symmetric operator such that hAv; vi D kvk2

(2)

for v 2 V;

(14.19)

B W V V ! V 0 is a bilinear, continuous operator such that hB.u; v/; vi D 0 for

v 2 V:

(14.20)

Lemma 14.1. Given > 0 and s 2 R there exists a smooth function w 2 H 1 .˝/2 such that div w D 0 in ˝, w D 0 on D , w is L-periodic in x1 , w D .s; 0/, wn D 0 on C , and for all v 2 V jhB.v; w/; vij kvk2 :

(14.21)

Proof. Let w be of the form w.x2 / D .s.x2 =h0 /; 0/, where W Œ0; 1/ ! Œ0; 1 is a smooth function such that .0/ D 1, 0 .0/ D 0, supp Œ0; minf 2jsj ; h0 ; 1/g, and

14.3 Existence of Global in Time Solutions

323

h0 is the minimum value of h.x1 / on Œ0; L. It is clear that all the stated properties of w other than (14.21) hold. To prove (14.21) observe that under our assumptions Z

jrot wj2 dx D ˝

Z

jrwj2 dx;

(14.22)

˝

and then ˇ ˇ ˇvˇ jhB.v; w/; vij jrvjkx2 w.x2 /kL1 .Œ0;h0 / ˇˇ ˇˇ x2

(14.23)

kvk 2kvk D kvk2 2 t u

in view of the Hardy inequality.

Lemma 14.2. Let j W R ! R satisfy .j1/–.j3/. For any solution v of Problem 14.1, 3

kv 0 .t/kV 0 C3 .1 C kv.t/k C jv.t/j1=2 kv.t/k 2 /

for a.e.

t 2 RC ; (14.24)

with a constant C3 > 0 independent of v. Proof. Let v be a solution of Problem 14.1. For any test function z 2 V we have, for a.e. t 2 RC , jhv 0 .t/; zij kFkV 0 kzk C k.t/kL2 .C / k kkzk C jhG.v.t// Av.t/ BŒv.t/; zij: From the growth condition .j2/ it follows that k.t/kL2 .C / C.1 C kv.t/k/ with a constant C > 0. Moreover, jhAv.t/; zij C jhG.v.t//; zij Ckv.t/kkzk. It remains to estimate the nonlinear term. For all w 2 V we have (14.22). Now, from the Hölder and Ladyzhenskaya inequalities we obtain Z jhBŒv.t/; zij ˝

jrot v.t/jjv.t/jjzj dx kv.t/kkv.t/kL4 .˝/2 kzkL4 .˝/2

Cjv.t/j1=2 kv.t/k3=2 kzk: t u

In this way we obtain (14.24).

Theorem 14.1. Let the potential j satisfy .j1/–.j3/, F 2 V 0 . Then for every v0 2 H there exists v 2 K C such that v.0/ D v0 . R1 Proof. Let % 2 C01 ..1; 1// be a mollifier such that 1 %.s/ ds D 1 and %.s/ 0. %k W R ! R by %k .s/ D k%.ks/ for k 2 N and s 2 R. Then supp %k We1define k ; 1k . We consider jk W R ! R defined by the convolution Z jk .r/ D

R

%k .s/j.r s/ ds

for

r 2 R:

324


By assertion .A/ of Theorem 3.21 we have jk 2 C1 .R/. Let s 2 R. We calculate j0k .s/s

jk .s C hs/ jk .s/ D lim D lim C h h!0 h!0C

Z supp %k

%k .r/

j.s C hs r/ j.s r/ dr: h

Estimating the integrand from below, by the Lebourg mean value theorem (cf. Theorem 3.20) we get, for 2 .0; 1/ dependent on s; r; h, and 2 @j.s r C hs/, j.s C hs r/ j.s r/ D %k .r/.s r C hs C r hs/ h %k .r/ d1 d2 js r C hsj2 .c1 C c2 js r C hsj/.jrj C hjsj/

%k .r/ d1 d2 .jsj C 1 C hjsj/2 .c1 C c2 .jsj C 1 C hjsj// .1 C hjsj/ ;

%k .r/

where we used the fact that jrj zero and noting that k 1 we get

1 k

1. After integrating over r, passing with h to

j0k .s/s d1 d2 .jsj C 1/2 c1 c2 .jsj C 1/ de1 de2 jsj2 ; whence regularized functions jk still satisfy .j3/, where the constants d1 ; d2 are different than the ones for j, but independent on k, and still we have de2 2 0; kk 2 . Let us furthermore take the sequence Vk of finite dimensional spaces such that Vk is spanned by the first k eigenfunctions of the Stokes operator with the Dirichlet and periodic boundary conditions given in the definition of the space V. Then fVk g1 kD1 S approximate V from inside, i.e., 1 V D V. Moreover we take the sequence k kD1 v0k ! v0 strongly in H such that v0k 2 Vk . We formulate the regularized Galerkin problems for k 2 N: Find a continuous function vk W RC ! Vk such that for a.e. t 2 RC vk is differentiable and hvk0 .t/ C Avk .t/ C BŒvk .t/; zi C .j0k ..vk /n .; t//; zn /L2 .C /

(14.25)

D hF C G.vk .t//; zi; vk .0/ D v0k

(14.26)

for a.e. t 2 RC and for all z 2 Vk . We first show that if vk solves (14.25) then an estimate analogous to (14.17) holds.


325

We take z D vk .t/ in (14.25) and, using (14.19) and (14.20) as well as the fact that hB.v; w/; vi kvk2 for all v 2 V, where can be made arbitrarily small (Lemma 14.1), we obtain 1d jvk .t/j2 C kvk .t/k2 C .j0k ..vk /n .; t//; .vk /n .t//L2 .C / kFkV 0 kvk .t/k 2 dt C kvk .t/k2 for a.e. t 2 RC . Using .j3/ and the Cauchy inequality with some " > 0 we obtain 1d jvk .t/j2 C kvk .t/k2 C de1 m1 .C / de2 k.vk /n .t/k2L2 .C / C."/kFk2V 0 2 dt C . C "/kvk .t/k2 for a.e. t 2 RC , where " > 0 is arbitrary and the constant C."/ is positive. Note that by the trace theorem k.vk /n .t/k2L2 .C / kvk .t/k2L2 .C /2 k k2 kvk .t/k2 . It follows that 1d jvk .t/j2 C . de2 k k2 /kvk .t/k2 C."/kFk2V 0 C . C "/kvk .t/k2 C jde1 jL 2 dt for a.e. t 2 RC . It is enough to take D " D independent of t,

e d2 kk2 . 4

We get, with C1 ; C2 > 0

1d jvk .t/j2 C C1 kvk .t/k2 C2 .1 C kFk2V 0 /: 2 dt Note that, after integration, Z t 1 1 jvk .t/j2 C C1 kvk .s/k2 ds jv0k j2 C C2 .1 C kFk2V 0 /t 2 2 0

(14.27)

for all

t 0: (14.28)

Existence of the solution to the Galerkin problem (14.25)–(14.26) is standard and follows by the Carathéodory theorem and estimate (14.28). Note that for all k 2 N solutions of (14.25) satisfy the estimate of Lemma 14.2, where the constants do not depend on initial conditions and the dimension k. We deduce from (14.27) that for all T > 0 vk

is bounded in

L2 .0; TI V/ \ L1 .0; TI H/:

(14.29)

We denote by W V ! L2 .C / the linear and bounded map given as v D . v/n and by W L2 .C / ! V 0 its adjoint operator. Now if Pk is the projection operator onto the space Vk , (14.25) is equivalent to the following equation in V 0 , vk0 .t/ C Avk .t/ C Pk BŒvk .t/ C Pk j0k ..vk /k .; t// D Pk F C Pk G.vk .t//:

326


From the choice of the space Vk it follows by Theorem 3.17 that kPk ukV 0 kukV 0 and by the argument of Lemma 14.2 it follows that for all T > 0 4

vk0

is bounded in L 3 .0; TI V 0 /:

(14.30)

In view of (14.29) and (14.30), by diagonalization, we can construct a subsequence such that vk ! v

2 weakly in Lloc .RC I V/

(14.31)

and vk0 ! v 0

4

3 weakly in Lloc .RC I V 0 /:

(14.32)

First we show that v.0/ D v0 in H. To this end choose T > 0. We have, for t 2 .0; T/, Z vk .t/ D vk0 C

t 0

Z

vk0 .s/ ds;

v.t/ D v.0/ C

t 0

v 0 .s/ ds;

where the equalities hold in V 0 . Take 2 V. Then Z 0

T

hvk .t/ v.t/; i dt Z

T

D 0

Z hvk0 v.0/; i dt C Z

D Thvk0 v.0/; i C

0

T

0

T

Z 0

t

.vk0 .s/ v 0 .s// ds; dt

hvk0 .t/ v 0 .t/; .T t/i dt:

The left-hand side of this equality goes to zero with k ! 1 as vk ! v weakly in L2 .0; TI V/. The last integral on the right-hand side also goes to zero with k ! 1 4 as vk0 ! v 0 weakly in L 3 .0; TI V 0 / and .T t/ 2 L4 .0; TI V/. Hence, vk0 ! v0 weakly in V 0 and v.0/ D v0 in H. Now we pass with k ! 1 in Eq. (14.25). To this end let us choose T > 0. We multiply (14.25) by 2 C.Œ0; T/ and integrate with respect to t in Œ0; T. Passing to the limit in linear terms A and G is standard. We focus on the multivalued term and the convective term. Let Z D H 1ı .˝/2 \ V for ı 2 0; 12 be endowed with H 1ı norm. Since V Z and Z V 0 , with compact and continuous embeddings, respectively, from the Aubin–Lions compactness theorem it follows that, for a subsequence, vk ! v strongly in L2 .0; TI Z/, and, by the continuity of the trace operator and also of the corresponding Nemytskii operator, we have for all T > 0, .vk /n ! vn

strongly in L2 .0; TI L2 .C //:

(14.33)


327

By assertion .B/ of Theorem 3.21, the functions j0k satisfy the growth condition .j2/ with the constants independent of k. It follows that the sequence j0k ..vk /n .; // is bounded in L2 .0; TI L2 .C //, and we can extract a subsequence, not renumbered, such that for all T > 0 j0k ..vk /n .; // !

weakly in L2 .0; TI L2 .C //:

(14.34)

2 We need to show that .t/ 2 [email protected] for a.e. t 2 RC . We use assertion .C/ of n .;t// Theorem 3.21, whence we have, for all T > 0,

.x; t/ 2 @j.vn .x; t//

.x; t/ 2 C .0; T/;

for a.e.

and the desired result follows. Hence we can pass to the limit in the multivalued term. Now we show the convergence in the nonlinear term, namely, that (for a subsequence) Z TZ Z TZ .rot vk .t/ vk .t// z.x/.t/ dx dt ! .rot v.t/ v.t// z.x/.t/ dx dt: 0

0

˝

˝

(14.35)

First we prove that Z

T

Z

0

˝

.rot vk .t/ vk .t// z.x/.t/ dx dt D Z 0

T

Z ˝

1 2

Z

T

Z

0

C

.vk /2n .x; t/zn .x/.t/ dS dt

.vk .x; t/ r/z.x/vk .x; t/.t/ dx dt:

(14.36)

From (14.15), as well as the formulas .a b/ c D .b c/ a, and r F D 0; r G D 0 we have

H)

rot .F G/ D .G r/F .F r/G

(14.37)

Z

hBŒvk .t/; zi D

˝

.vk .t/ z/ rot vk .t/ dx

(14.38) Z

Z D

˝

rot.vk .t/ z/ vk .t/ dx C

@˝

.vk .t/ .vk .t/ z// n dS:

The surface integral is equal to zero and hence, using (14.37) in the right-hand side, we obtain Z Z hBŒvk .t/; zi D .z r/vk .t/ vk .t/dx .vk .t/ r/z vk .t/ dx: (14.39) ˝

˝

Integration by parts in the first integral on the right-hand side and then integration in the time variable give the result.

328


Since we can deal with the second term on the right-hand side of (14.36) as in the usual Navier–Stokes theory [219], we consider only the surface integral. Taking the difference of the corresponding terms and setting zn .x/.t/ D .x; t/ we obtain ?Z ? ?

0

T

Z C

? ? ..vk /n .x; t/ vn .x; t//..vk /n .x; t/ C vn .x; t// .x; t/ dS dt?

k.vk /n vn kL2 .0;TIL2 .C // kvk C vkL2 .0;TIL4 .C /2 / max k .t/kL4 .C / : t2Œ0;T

This proves (14.35) since the embedding H 1=2 .C /2 Lr .C /2 is continuous for r 1, vk is bounded in L2 .0; TI L4 .C /2 /, and k.vk /n vn kL2 .0;TIL2 .C // ! 0 by (14.33). Hence, the limit v solves Problem 14.1. It remains to show (14.18). The proof follows the lines of that of Theorem 8.1 in [61]. Let us fix 0 t1 < t2 and choose 2 C01 ..t1 ; t2 // with 0. We multiply (14.27) by and integrate by parts. We have Z t2 Z t2 Z 1 t2 jvk .t/j2 0 .t/ dt C C1 kvk .t/k2 .t/ dt C2 .1 C kFk2V 0 / .t/ dt: 2 t1 t1 t1 (14.40) We need to pass to the limit (k ! 1) in the last inequality. From (14.31) and (14.32) by the Aubin–Lions compactness theorem we conclude that vk ! v strongly in L2 .t1 ; t2 I H/. From inequality Z

t2

Z

2

t2

.jvk .t/j jv.t/j/ dt

t1

jvk .t/ v.t/j2 ds

t1

it follows that jvk .t/j ! jv.t/j strongly in L2 .t1 ; t2 /, and hence, for a subsequence, jvk .t/j2 ! jv.t/j2 for almost every t 2 .t1 ; t2 /. Since from (14.28) it follows that functions jvk .t/j2 0 .t/ have an integrable majorant on .t1 ; t2 /, we have, by the Lebesgue dominated convergence theorem that Z

t2

jvk .t/j2

0

Z .t/ dt !

t1

t2

jv.t/j2

0

.t/ dt:

(14.41)

t1 1

1

Now, from (14.31) it follows that vk .t/. .t// 2 ! v.t/. .t// 2 weakly in L2 .t1 ; t2 I V/ and hence, by the sequential weak lower semicontinuity of the norm we obtain Z t2 Z t2 kv.t/k2 .t/ dt lim inf kvk .t/k2 .t/ dt: (14.42) t1

k!1

t1

Thereby, we can pass to the limit in (14.40) which gives us (14.18) and the proof is complete. t u

14.4 Existence of Attractors

329

14.4 Existence of Attractors In this section we prove the existence of a trajectory attractor and a weak global attractor for the considered problem. We have shown that for any initial condition v0 2 H Problem 14.1 has at least one solution v 2 KC . The key idea behind the trajectory attractor (see [61, 62, 226, 227]) is, in contrast to the direct study of the solutions asymptotic behavior, the investigation of the family of shift operators fT.t/gt0 defined on K C by the formula .T.t/v/.s/ D v.s C t/: Before we pass to a theorem on the existence of the trajectory attractor, which is the main result of this section, we recall some definitions and results (see [61, 62, 226, 227]). Let 4

F .0; T/ D fu 2 L2 .0; TI V/ \ L1 .0; TI H/ W u0 2 L 3 .0; TI V 0 /g; and 4

2 1 3 FCloc D fu 2 Lloc .RC I V/ \ Lloc .RC I H/ W u0 2 Lloc .RC I V 0 /g:

Note that F .0; T/ Cw .Œ0; TI H/ and from the Lions–Magenes lemma (cf. Lemma 1.4, Chap. 3 in [219], Theorem II,1.7 in [62] or Lemma 2.1 in [227]) it follows that for all u 2 F .0; T/ we have ju.t/j kukL1 .0;TIH/

for all

t 2 Œ0; T:

(14.43)

Furthermore, let us define the Banach space FCb D fu 2 FCloc W kukF b < 1g; C

where the norm in FCb is given by kukF b D sup k˘0;1 u. C h/kF .0;1/ : C

h0

In the above definition ˘0;1 u./ is a restriction of u./ to the interval .0; 1/, and jjvjjF .0;1/ D kvkL2 .0;1IV/ C jjvjjL1 .0;1IH/ C kv 0 k

4

L 3 .0;1IV 0 /

:

loc Finally, we define the topology by C in the space FCloc in the following way: the loc loc 1 if sequence fuk gkD1 FC is said to converge to u 2 FCloc in the sense of C

330


uk ! u

weakly in

2 Lloc .RC I V/;

uk ! u

weakly- in

1 Lloc .RC I H/;

u0k ! u0

weakly in

3 Lloc .RC I V 0 /:

4

loc is stronger than the topology Note (see, for example, [227]) that the topology C loc of Cw .Œ0; TI H/ for all T 0 and hence from the fact that vk ! v in C it follows that vk .t/ ! v.t/ weakly in H for all t 0.

Definition 14.1. A set P K C is said to be absorbing for the shift semigroup fT.t/gt0 if for any set B K C bounded in FCb there exists D .B/ > 0 such that T.t/B P for all t . Definition 14.2. A set P K C is said to be attracting for the shift semigroup loc fT.t/gt0 in the topology C if for any set B K C bounded in FCb and for any loc neighborhood O.P/ of P in the topology C there exists D .B; O.P// > 0 such that T.t/B O.P/ for all t . Definition 14.3. A set U K C is called a trajectory attractor of the shift loc semigroup fT.t/gt0 on K C in the topology C if loc (a) U is bounded in FCb and compact in the topology C , loc (b) U is an attracting set in the topology C , (c) U is invariant, i.e., T.t/U D U for any t 0.

In order to show the existence of a trajectory attractor for Problem 14.1 we will use the following theorem (see, for instance, Theorem 4.1 in [227]). Theorem 14.2. Assume that the trajectory set K and that T.t/K

C

C

is contained in the space FCb

K C:

(14.44)

Suppose that the semigroup fT.t/gt0 has an attracting set P that is bounded in the loc norm of FCb and compact in the topology C . Then the shift semigroup fT.t/gt0 has a trajectory attractor U P. We are in position to formulate the main theorem of this section. Theorem 14.3. The shift semigroup fT.t/gt0 defined on the set K C of solutions of Problem 14.1 has a trajectory attractor U which is bounded in FCb and compact loc in the topology C . Before we pass to the proof of this theorem we will need three auxiliary lemmas. Lemma 14.3. Let v 2 K C . For all h 0 the following estimates hold: kvk2L2 .h;hC1IV/ C kvk2L1 .h;hC1IH/ C4 C C5 kvk2L1 .0;1IH/ eC6 h ;

(14.45)

14.4 Existence of Attractors 4

kv 0 k 3 4

L 3 .h;hC1IV 0 /

331 8

C7 C C8 kvkL31 .0;1IH/ eC9 h ;

(14.46)

where C4 ; C5 ; C6 ; C7 ; C8 , and C9 are positive constants independent of h; v. Proof. Let us fix h 0 and v 2 K C . Since for all u 2 V, 1 juj2 kuk2 with the constant 1 > 0, it follows from (14.18), that for all 0 t1 < t2 and 2 C01 ..t1 ; t2 // with 0, Z t2 Z t2 Z 1 t2 2 0 2 2 jv.t/j .t/ dt C C1 1 jv.t/j .t/ dt C2 .1 C kFkV 0 / .t/ dt: 2 t1 t1 t1 (14.47) We are in position to use Lemma 9.2 from [61] to deduce that there exists a Lebesgue null set Q RC such that for all t; 2 RC n Q we have jv.t/j2 eC1 1 .t / jv./j2

eC1 1 .t / 1 C2 .1 C kFk2V 0 /: C1 1

(14.48)

Now, since the function t ! v.t/ is weakly continuous, it follows that t ! jv.t/j2 is lower semicontinuous and (14.48) holds for all t (but only for a.e. 2 RC ). Hence, for all t > 0 we can find 2 .0; minf1; tg/ such that jv.t/j2 eC1 1 .t / kvk2L1 .0;1IH/ C

eC1 1 .t / C2 .1 C kFk2V 0 /; C1 1

(14.49)

C2 .1 C kFk2V 0 / : C1 1

(14.50)

and moreover, for all t > 0, we have jv.t/j2 eC1 1 .1t/ kvk2L1 .0;1IH/ C

By Corollary 9.2 in [61] it follows from (14.18) that for almost all h > 0, 1 jv.h C 1/j2 C C1 2

Z

hC1 h

1 kv.t/k2 dt C2 .1 C kFk2V 0 / C jv.h/j2 : 2

Using (14.50) we obtain, for a.e. h > 0, Z

hC1 h

kvk2L1 .0;1IH/ eC1 1 .1h/ C2 .1 C kFk2V 0 / C2 2 kv.t/k dt .1 C kFkV 0 / C C ; C1 2C1 2C12 1 2

and the estimate (14.45) follows for all h 0 since both left- and right-hand side of the above inequality are continuous functions of h. Now, using (14.24) we get

4 2 kv 0 .t/kV3 0 C10 1 C 1 C jv.t/j 3 kv.t/k2

for a.e.

t 2 RC ;

332


where C10 > 0. Integrating this inequality from h to h C 1 we obtain Z

hC1 h

Z 2 4=3 kv 0 .t/kV 0 dt C10 C C10 1 C jjv.t/jjL31 .h;hC1IH/

hC1

kv.t/k2 dt:

h

Inequality (14.46) follows from an application of (14.45). Lemma 14.4. The set K

C

is closed in the topology

Proof. Assume that for a sequence fvk g1 kD1 K

C

t u

loc C .

we have

vk ! v

weakly in

2 Lloc .RC I V/;

vk ! v

weakly- in

1 Lloc .RC I H/;

vk0 ! v 0

weakly in

4

3 Lloc .RC I V /:

C We need to show that v satisfies (14.16) and (14.18). Since fvk g1 kD1 K , we have, for all k 2 N and z 2 V,

hvk0 .t/ C Avk .t/ C BŒvk .t/; zi C .k .t/; zn /L2 .C / D hF; zi C hG.vk .t//; zi; 2 k .t/ 2 [email protected] k /n .;t//

for a.e. t 2 RC . Passing to the limit in terms with A; B; G is analogous to that in the proof of Theorem 14.1 (see also [62, 176, 219, 220]). In order to pass to the limit in the multivalued term observe that from .j2/ it follows that for a.e. t 2 RC kk .t/k2L2 .C / 2c21 m1 .C / C 2c2 k k2 kvk .t/k2 : Hence, after integration in the time variable we get, for all T 2 RC , kk k2L2 .0;TIL2 .C // 2Tc21 L C 2c2 k k2 kvk k2L2 .0;TIV/ : 2 By a diagonal argument it follows that there exists 2 Lloc .RC I L2 .C // such that, for a subsequence,

k !

2 weakly in Lloc .RC I L2 .C //:

As in the proof of Theorem 14.1, .vk /n ! vn

2 strongly in Lloc .RC I L2 .C //:

Using assertion .C/ of Theorem 3.21 we deduce that .x; t/ 2 @j.vn .x; t//

a.e. in C RC ;

14.4 Existence of Attractors

333

and moreover, 2 .t/ 2 [email protected] n .;t//

a.e. in RC :

Hence we can pass to the limit in the term .k .t/; zn /L2 .C / . It remains to show that v satisfies (14.18). The argument is similar to that in the proof of Theorem 14.1. We choose 0 t1 < t2 and 2 C01 ..t1 ; t2 //. We have, after possible refining to 2 0 a subsequence, jvk .t/j .t/ ! jv.t/j2 0 .t/ for a.e. t 2 .t1 ; t2 /. Moreover, since weakly-* convergent sequences in L1 .t1 ; t2 I H/ are bounded in this space it follows that there exists a majorant for jvk .t/j2 0 .t/. Hence, by the Lebesgue dominated 1 1 convergence theorem and since vk .t/. .t// 2 ! v.t/. .t// 2 weakly in L2 .t1 ; t2 I V/, we can pass to the limit in (14.18) written for vk , which completes the proof. t u Proof (of Theorem 14.3). Observe that from Lemma 14.3 it follows that for every v 2 K C , kvkF b < 1. Note that since the problem is autonomous, for any v 2 C

K C and any h 2 RC , the function vh , defined as vh .t/ D v.t C h/ for all t 0, belongs to K C and hence (14.44) holds. Now let us estimate kT.s/vkF b for v 2 K C and s 0. We have C

kT.s/vkF b

C

n D sup kvkL2 .h;hC1IV/ C jjv.t/jjL1 .h;hC1IH/ C kv 0 k

4 L3

hs

o .h;hC1IV 0 /

:

Using Lemma 14.3 we get ( 34 ) 1

8 2 C6 h 2 C h kT.s/vkF b sup C 2 1 C kvkL1 .0;1IH/ e C 1 C kvkL31 .0;1IH/ e 9 ; C

hs

with a constant C > 0 dependent only on C4 ; C5 ; C7 ; C8 of Lemma 14.3. By a simple calculation we obtain, for all s 2 RC , ˇ

ˇ

kT.s/vkF b R0 C CkvkL1 .0;1IH/ eıs R0 C CkvkF .0;1/ eıs ; C

(14.51)

where C; R0 ; ˇ; ı > 0 do not depend on s; v. Now we define the set P as P D fv 2 K

C

W kvkF b 2R0 g: C

We will show that P is absorbing (and hence also attracting) for fT.t/gt0 . Let B be bounded in FCb . Hence B is bounded in F .0; 1/. Let kvkF .0;1/ R for v 2 B. Now we choose s0 > 0 such that CRˇ eıs0 R0 . From (14.51) we have for s s0 that kT.s/vkF b 2R0 ; C

and we deduce that P is absorbing.

334


loc It suffices to show that P is compact in the topology C . The fact that it is relatively compact follows from its boundedness and basic properties of weak compactness in reflexive Banach spaces. It remains to show that it is closed. Let loc C

vk ! v and fvk g P. From the weak lower semicontinuity of the norm it follows that kvkF b lim inf kvk kF b 2R0 : C

k!1

C

Moreover from Lemma 14.4 it follows that v 2 K C . Thus v 2 P and the proof is complete. t u Now we show that any section of the trajectory attractor is a weak global attractor. We will start from the definition of a weak global attractor (cf., e.g., [227]). Definition 14.4. A set A H is called a weak global attractor of the set K is weakly compact in H and the following properties hold:

C

if it

.i/ for any set B K C bounded in the norm of FCb its section B.t/ is attracted to A in the weak topology of H as t ! 1, that is, for any neighborhood Ow .A / of A in the weak topology of H there exists a time D .B; Ow .A // such that B.t/ Ow .A /

for all

t ;

.ii/ A is the minimal weakly closed set in H that attracts the sections of all bounded sets in K C in the weak topology of H as t ! 1.

Fig. 14.2 Schematic illustration of trajectory attractor for the semiflow governed by the solution map of the one-dimensional ODE xP D jxj.1 x/. The trajectory attractor U consists of two constant trajectories located at the stationary points 0 and 1, and of the family of bounded complete trajectories that connect those points. The global attractor A is the section of the trajectory attractor A D U.0/ D U.t/ for all t 2 R


335

We prove the following Theorem. Theorem 14.4. The set A D U.0/ H is a weak global attractor of K C . Proof. The argument is standard in the theory of trajectory attractors and follows the lines of the proof of Assertion 5.3 in [227]. Since U is bounded in FCb and thus in F .0; 1/, by (14.43) we deduce that U.0/ is bounded in H. Moreover, since the loc topology C is stronger than the topology of Cw .Œ0; TI H/ it follows that if uk ! u loc in the topology C then uk .0/ ! u.0/ weakly in H. From the fact that U is compact it follows that U.0/ is weakly compact in H. To show .i/ let us take B K C bounded in the norm of FCb . This set is loc attracted to U in the topology C . Since for every sequence uk ! u in the topology loc C we have uk .0/ ! u.0/ weakly in H, it follows that .T.t/B/.0/ is attracted to U.0/ in the weak topology of H, or, in other words, B.t/ is attracted to U.0/ in the weak topology of H. To show .ii/ let E be an arbitrary weakly closed subset of H that weakly attracts the sections of bounded sets B K C , that is, for any weak neighborhood Ow .E /, B.t/ Ow .E / for all t D .B; Ow .E //. Let us take B D U. We have U.t/ Ow .E / for all t D .U; Ow .E //. From the invariance of the trajectory attractor we obtain that A D U.0/ D U.t/ Ow .E /. Since E is weakly closed in H, it follows that A E and the proof is complete. t u The example which illustrates the fact that the global attractor is the section of the trajectory attractor is presented in Fig. 14.2.

14.5 Comments and Bibliographical Notes In this chapter we followed [124]. The system of Eqs. (14.1)–(14.2) with no-slip boundary conditions: (14.3) at D for h D const and u D const on C (instead of (14.4)–(14.5) on C ) was intensively studied in several contexts, some of them mentioned in the introduction of Boukrouche and Łukaszewicz [26]. The autonomous case with h ¤ const and with u D const on C was considered in [22, 24] (also see Chap. 9) and the nonautonomous case h ¤ const, u D U.t/e1 on C was considered in [28] (also see Chap. 13). Existence of exponential attractors for the Navier–Stokes fluids and global attractors for Bingham fluids, with the Tresca boundary condition on C was proved in [162] (also see Chap. 10) and [27], respectively. Recently, attractors for two-dimensional Navier–Stokes flows with Dirichlet boundary conditions were studied in [79], where the time continuous problem has a unique solution but a solution of the semidiscrete problem constructed by the implicit Euler scheme can be nonunique, and hence the theory of multivalued semiflows is needed to study the time discretized systems. Asymptotic behavior of solutions for the problems governed by partial differential inclusions where the multivalued term has the form of Clarke subdifferential

336


was studied in [132, 133], where the reaction-diffusion problem with multivalued semilinear term was considered, and in [123], where the strongly damped wave equation with multivalued boundary conditions was analyzed. For the problem considered in this chapter, existence of weak solutions for the case u D 0 in place of (14.4) was shown in [176]. Our assumptions .j1/–.j3/ are more general than the corresponding assumptions of Theorem 1 in [176], save for the fact that j is assumed there to depend on space and time variables directly. To prove the existence of attractors we used the theory of trajectory attractors, which, instead of the direct analysis of the multivalued semiflow (i.e., map that assigns to initial condition the set of states obtainable after some time t, see [171, 239] or the review paper [10]), focuses on the shift operator defined on the space of trajectories for the studied problem. This approach was introduced in papers [60, 167, 210] as a method to avoid the nonuniqueness of solutions, indeed, the shift operator is uniquely defined even if the dynamics of the problem is governed by the multivalued semiflow. Recent results and open problems in the theory of trajectory attractors are discussed in the survey papers [10, 227].

15

Evolutionary Systems and the Navier–Stokes Equations

This chapter is devoted to the study of three-dimensional nonstationary Navier–Stokes equations with the multivalued frictional boundary condition. We use the formalism of evolutionary systems to prove the existence of weak global attractor for the studied problem.

15.1 Evolutionary Systems and Their Attractors We remind the framework of evolutionary systems [65, 66]. Let .X; ds .; // be a metric space. The metric ds will be referred to as the strong metric. We denote by dw another metric on X, referred to as the weak metric, satisfying the following conditions: 1. the space X is dw compact, 2. if limk!1 ds .uk ; vk / D 0 for some sequences uk ; vk limk!1 dw .uk ; vk / D 0.

2

X, then also

Any strongly compact (i.e., ds compact) subset of X is also weakly compact (i.e., dw compact). Any strongly closed (i.e., ds closed) subset of X is also weakly closed (i.e., dw closed) and, as a subset of the dw compact set X, also dw compact. To define the evolutionary system, let T D fI W I D ŒT; 1/ R

or

I D .1; 1/g:

By F .I/ we denote the set of all X-valued functions having the domain I 2 T .


337

338

15 Evolutionary Systems and the Navier–Stokes Equations

Definition 15.1. A map E that associates to each I 2 T a subset E .I/ F .I/ is called an evolutionary system if the following conditions hold: .i/ .ii/ .iii/ .iv/

E .Œ0; 1// ¤ ;, E .I C s/ D fu./ W u. s/ 2 E .I/g for all s 2 R, fu./jI2 W u./ 2 E .I1 /g E .I2 / for all I1 ; I2 2 T such that I2 I1 , E ..1; 1// D fu./ W u./jŒT;1/ 2 E .ŒT; 1// for all T 2 Rg.

For I D ŒT; 1/ the set E .I/ is referred to as the set of trajectories on the interval I, and E ..1; 1// is called the set of complete trajectories. Having the evolutionary system we can define the map G W Œ0; 1/ X ! 2X by G .t; x/ D fu.t/ W u.0/ D x

for some u 2 E .Œ0; 1//g:

We can naturally extend G to Œ0; 1/ 2X by defining G .t; A/ D

S x2A

G .t; x/.

Remark 15.1. If we additionally assume the existence, i.e., that for any x0 2 X there exists u 2 E .Œ0; 1// such that u.0/ D x0 , then the map G becomes an m-semiflow in the sense of Melnik and Valero [171], that is: .i/ .ii/

G .0; x/ D fxg for all x 2 X, G .t C s; X/ G .t; G .s; x// for all x 2 X and s; t 0.

The illustration of an evolutionary system is presented in Fig. 15.1.

Fig. 15.1 Illustration of an evolutionary system. If a semiflow (or m-semiflow) is dissipative, i.e., has a bounded absorbing set X, then this set, endowed with appropriate metrics becomes the metric space for the evolutionary system. Moreover, the endings of all trajectories, which must be contained in the absorbing set (thickened line in the figure), constitute the family E .I/

Let A X and r 0. We denote Bw .A; r/ D fx 2 X W distw .x; A/ < rg, where distw .x; A/ D infy2A dw .x; y/. Moreover, for sets A; B X we define the weak Hausdorff semi-distance distw .A; B/ D supx2A distw .x; B/. We pass to the definition of uniformly weakly attracting sets and weak attractors. Definition 15.2. A set A X is uniformly weakly attracting (uniformly dw attracting) if for any " > 0 there exists t0 > 0 such that G .t; X/ Bw .A; "/ for all t t0 , or, equivalently lim distw .G .t; X/; A/ D 0:

t!1

15.1 Evolutionary Systems and Their Attractors

339

Definition 15.3. A set A X is a weak global attractor (dw global attractor) if it is a minimal weakly closed uniformly weakly attracting set. Of course, the weak global attractor must be weakly compact. The following theorem gives the existence of the weak global attractor. Theorem 15.1 (cf. [65, Theorem 3.9]). Every evolutionary system possesses a weak global attractor A X. Let a; b 2 R be such that a < b. Consider the Banach space Cw .Œa; bI X/ of weakly continuous functions from the interval Œa; b to the space X. This space is a metric one, equipped with the metric dCw .Œa;bIX/ .u; v/ D sup dw .u.t/; v.t//: t2Œa;b

This metric can be naturally extended to Cw .Œa; 1/I X/ by the formula dCw .Œa;1/IX/ .u; v/ D

X 1 supt2Œa;aCk dw .u.t/; v.t// : 2k 1 C supt2Œa;aCk dw .u.t/; v.t// k2N

We make the following assumption on the evolution system: .A1/

the set E .Œ0; 1// is compact in Cw .Œ0; 1/I X/:

Assumption .A1/ will yield the invariance of the global attractor. To define the notion of invariance we first set GQ.t; A/ D fu.t/ W u.0/ 2 A for

u 2 E ..1; 1//g;

where

A X; t 2 R:

Definition 15.4. The set A X is said to be invariant (in the sense of Foia¸s– Cheskidov) if GQ.t; A/ D A for all t 0. Remark 15.2. Since GQ.t; A/ G .t; A/, then the fact that A is invariant implies that A G .t; A/, i.e., A is negatively semi-invariant in the sense of Melnik and Valero (see [171]). Theorem 15.2 (cf. [65, Theorem 5.8]). Let E be an evolutionary system that satisfies .A1/. Then the weak global attractor A is the maximal invariant set. Moreover, A D fx 2 X W x D u.0/

for some

u 2 E ..1; 1//g;

where, by invariance, we can also take values at any t 0 instead of zero. Moreover, for any " > 0 there exists t0 > 0 such that for any t > t0 and for any trajectory u 2 E .Œ0; 1// we have dCw .Œt ;1/IX/ .u; v/ < " for a certain complete trajectory v 2 E ..1; 1//. The last assertion of the above theorem is known as the tracking property.

340


15.2 Three-Dimensional Navier–Stokes Problem with Multivalued Friction We will consider a flow between two surfaces, the flat one at the bottom, and the periodic one at the top. Define ˝1 D f.x1 ; x2 ; x3 / 2 R3 W 0 < x3 < h.x1 ; x2 /g, where the function h.; x2 / is assumed to be L1 -periodic, and h.x1 ; / is assumed to be L2 -periodic. Moreover we assume that h is Lipschitz and that min

.x1 ;x2 /2Œ0;L1 Œ0;L2

h.x1 ; x2 / > 0:

The periodicity of h implies that it suffices to consider the flow inside the domain ˝ D f.x1 ; x2 ; x3 / 2 R3 W .x1 ; x2 / 2 .0; L1 / .0; L2 /; 0 < x3 < h.x; y/g: Let t0 2 R be the initial time. The momentum equation and the incompressibility condition are, respectively, given by ut u C .u r/u C rp D f div u D 0 on

on

˝ .t0 ; 1/;

˝ .t0 ; 1/;

(15.1) (15.2)

where we look for the velocity u W ˝.t0 ; 1/ ! R3 and pressure p W ˝.t0 ; 1/ ! R. The function f W ˝ ! R3 is the mass force density, and > 0 is the viscosity coefficient. The boundary of ˝ is divided into the following parts: D D .0; L1 / .0; L2 / f0g; C D f.x1 ; x2 ; x3 / 2 R3 W .x1 ; x2 / 2 .0; L1 / .0; L2 /; x3 D h.x1 ; x2 /g; L D L1 [ L2 ; L1 D f0; L1 g f.x2 ; x3 / 2 R2 W 0 < x2 < L2 ; 0 < x3 < h.0; x2 /g; L2 D f.x1 ; x2 ; x3 / 2 R3 W 0 < x1 < L1 ; x2 2 f0; L2 g; 0 < x3 < h.x1 ; 0/g: The domain ˝ and three parts of its boundary are schematically presented in Fig. 15.2. The stress tensor T is associated with the symmetrized displacement gradient D.u/ D 12 .ru C ru| / and the pressure by the linear constitutive law T D pI C 2D.u/. On the boundary of ˝ the stress can be decomposed into normal and tangential components: Tn D .Tn/ n and T D Tn Tn n. On D we assume the homogeneous Dirichlet boundary condition uD0

on

D .t0 ; 1/:

(15.3)

15.2 Three-Dimensional Navier–Stokes Problem with Multivalued Friction

341

Fig. 15.2 The example setup of the studied problem. We assume the frictional contact between the flowing liquid and the foundation on C , periodic boundary conditions on L , and the homogeneous Dirichlet boundary conditions on D

We assume the periodicity of normal stresses and velocities on L , i.e., .x1 ; x2 ; x3 ; t/ 2 L1 Œt0 ; 1/

)

u.0; x2 ; x3 ; t/ D u.L1 ; x2 ; x3 ; t/

(15.4)

^ .Tn/.0; x2 ; x3 ; t/ D .Tn/.L1 ; x2 ; x3 ; t/; .x1 ; x2 ; x3 ; t/ 2 L2 Œt0 ; 1/

)

u.x1 ; 0; x3 ; t/ D u.x1 ; L2 ; x3 ; t/

(15.5)

^ .Tn/.x1 ; 0; x3 ; t/ D .Tn/.x1 ; L2 ; x3 ; t/: Note that, due to the periodicity of u, the periodicity of normal stresses on L1 means that, for .x1 ; x2 ; x3 / 2 L1 we have p.0; x2 ; x3 ; t/ D p.L1 ; x2 ; x3 ; t/ and ui;1 .0; x2 ; x3 ; t/ D ui;1 .L1 ; x2 ; x3 ; t/ for i ¤ 1; and on L2 we have p.x1 ; 0; x3 ; t/ D p.x1 ; L2 ; x3 ; t/ and ui;2 .x1 ; 0; x3 ; t/ D ui;2 .x1 ; L2 ; x3 ; t/ for i ¤ 2: Assume that the velocity u is sufficiently smooth. The fact that on L1 the function u.; x2 ; x3 / is L1 -periodic implies the same periodicity of partial derivatives ui;2 and ui;3 for i D 1; 2; 3, and, by the incompressibility condition, also of u1;1 . Similarly, from L2 -periodicity of u.x1 ; ; x3 / on L2 we obtain the same L2 -periodicity of the partial derivatives ui;1 and ui;3 for i D 1; 2; 3, and, by the incompressibility condition, also of u2;2 . On C we assume that un D 0

on C .t0 ; 1/;

(15.6)

T 2 g.u /

on C .t0 ; 1/;

(15.7)

where g W C R ! 2R is a friction multifunction, un D u n is the normal velocity, and u D u un n is the tangential velocity. Finally, we need the initial condition u D u0

on

˝

for t D t0 :

(15.8)

342


We pass to the weak formulation for the studied problem. First, we define the spaces: VQ Dfv 2 C1 .˝/3 W div v D 0 in ˝;

v D 0 on D ;

v.; x2 ; x3 / is L1 -periodic on L1 ; V D closure of VQ in H 1 .˝/3

and

vn D 0 on C ;

v.x1 ; ; x3 / is L2 -periodic on L2 g; H D closure of VQ in L2 .˝/3 :

The spaces V H V 0 constitute the evolution triple with dense and continuous embeddings. Moreover, the embedding V H is compact. As the Dirichlet 1 boundary is nontrivial, R the norm in the space V equivalent with the standard H 2 norm3 2 is given by kvk D ˝ rv.x/W rv.x/ dx. We will also use the space W D L .C / as well as the linear and continuous trace operator W V ! W, whose norm we will denote by k kL .VIW/ D k k. The calligraphic symbols will be used for timedependent spaces: V D L2 .t0 ; TI V/, V 0 D L2 .t0 ; TI V 0 /. To motivate the definition of the weak solution assume that smooth functions u; p satisfy (15.1)–(15.8). Multiplying the momentum equation (15.1) by v 2 VQ and integrating over ˝, after application of the incompressibility condition (15.2), the boundary conditions (15.3)–(15.7), and integration by parts, we arrive at the following weak form of the problem, Z

Z ˝

ut v dx C

Z ru rv dx C

˝

˝

Z .u r/u v dx C

C

Z v dS D

˝

f v dx;

where .x; t/ 2 g.x; u .x; t// for .x; t/ 2 C Œt0 ; 1/. Exercise 15.1. Derive the weak formulation of the considered problem. Hint. The derivation of the above weak form is similar to that in Chap. 5. Note that due to theR fact that the contact boundary is a flat surface R we can consider the bilinear form ˝ ru.x/W rv.x/ dx instead of more general ˝ D.u/.x/W D.v/.x/ dx (see Remark 5.1). Moreover, the periodic conditions on u; Tn, and the definition of the space VQ imply that all boundary integrals on L cancel.

15.3 Existence of Leray–Hopf Weak Solution We assume that .t0 ; T/, the time interval, is given. Using the Galerkin method we will establish the existence of the weak solution for considered problem, understood in the Leray–Hopf sense, on the interval .t0 ; T/. First, we define the operator A W V ! V 0 by the formula Z hAu; vi D

ru.x/W rv.x/ dx ˝

for

u; v 2 V:

15.3 Existence of Leray–Hopf Weak Solution

343

Obviously, the operator A is linear and bounded, moreover hAu; ui D kuk2 for u 2 V. We define the Nemytskii operator A W V ! V 0 by the formula .A u/.t/ D A.u.t// for a.e. t 2 .t0 ; T/. This Nemytskii operator is also linear and bounded, and hence it is continuous and moreover also weakly sequentially continuous, i.e., uk ! u weakly in V implies that A uk ! A u weakly in V 0 . Next, we define the bilinear operator B W V V ! V 0 as Z hB.u; w/; vi D

˝

.u r/w v dx:

Similar as in Lemmata 5.4 and 5.5 we have for u; v; w 2 V, hB.u; w/; vi cB kukkwkkvk;

(15.9)

hB.u; v/; vi D 0;

(15.10)

hB.u; w/; vi D hB.u; v/; wi;

(15.11)

and B is sequentially weakly continuous, i.e., wk ! w

and

uk ! u weakly in V )

(15.12)

hB.wk ; uk /; vi ! hB.w; u/; vi

for all

v 2 V:

4

The Nemytskii operator for B has values only in L 3 .t0 ; TI V 0 / and not in V 0 . Indeed, let us consider the function .t0 ; T/ 3 t ! B.u.t/; v.t// 2 V 0 , where u; v 2 V \ L1 .t0 ; TI H/. We have Z kB.u.t/; v.t//kV 0 D sup

Z

kwkD1 ˝

.u.t/ r/v.t/ w dt D sup

kwkD1 ˝

.u.t/ r/w v.t/ dt

ku.t/kL4 .˝/3 kv.t/kL4 .˝/3 :

(15.13)

We use the following Ladyzhenskaya interpolation inequality valid for all u 2 H 1 .˝/3 (see Theorem 3.11) 1

3

kukL4 .˝/3 CkukL42 .˝/3 kukH4 1 .˝/3 : In our case we have 1

3

kukL4 .˝/3 Cjuj 4 kuk 4 ;

(15.14)

and hence 4

1

1

kB.u.t/; v.t//kV3 0 C2 ju.t/j 3 ku.t/kjv.t/j 3 kv.t/k;

344


whereas we can define 4

B W .V \ L1 .t0 ; TI H// .V \ L1 .t0 ; TI H// ! L 3 .t0 ; TI V 0 / by .B.u; v//.t/ D B.u.t/; v.t//. We have the following result. Lemma 15.1. Let the sequences uk ! u and vk ! v both converge weakly in L2 .t0 ; TI V/ and strongly in L2 .t0 ; TI H/. Then we have B.uk ; vk / ! B.u; v/ weakly 4 in L 3 .t0 ; TI V 0 /. Proof. The proof follows the lines of the proof of Lemma III.3.2 in [219]. It is enough to prove that for smooth w W .t0 ; T/ ˝ ! R3 we have Z

T

Z

T

hB.uk .t/; vk .t//; w.t/i dt !

t0

hB.u.t/; v.t//; w.t/i dt:

t0

The result follows by the density of smooth functions in L4 .t0 ; TI V/, a predual space 4 of L 3 .t0 ; TI V 0 /. We have Z

T

hB.uk .t/; vk .t//; w.t/i dt D

t0

3 Z X

T

Z

i;jD1 t0

˝

.uk .t//i

@wj .t/ .vk .t//j dx dt: @xi

The last integrals converge to

3 Z X

T

i;jD1 t0

Z ˝

.u.t//i

@wj .t/ .v.t//j dx dt D @xi

Z

T

hB.u.t/; v.t//; w.t/i dt;

t0

t u

and the proof is complete. 0

We pick the initial condition u0 2 H and the source term f 2 V . 3 We assume that the friction multifunction g W C R3 ! 2R satisfies: .g1/ for every s 2 R3 the multifunction x ! g.x; s/ has a measurable selection, i.e., there exists gs W C ! R3 , a measurable function, such that gs .x/ 2 g.x; s/ for all x 2 C , .g2/ the set g.x; s/ is nonempty, compact, and convex in R3 , for all s 2 R3 and for a.e. x 2 C , .g3/ graph of s ! g.x; s/ is closed in R6 for a.e. x 2 C , .g4/ we have jj c1 C c2 jsj for all s 2 R3 , a.e. x 2 C and all 2 g.x; s/, with c2 0 and c1 2 R, 2 .g5/ we have the following dissipativity condition s d

1 d2 jsj for all s 2 R3 , a.e. x 2 C and all 2 g.x; s/ with d1 0 and d2 2 0; kk 2 . Finally, we define the multivalued operators N W W ! 2W and N W L2 .t0 ; TI W/ ! 2 2L .t0 ;TIW/ as


N.v/ Df 2 W W .x/ 2 g.x; v.x//

345

a.e. on C g;

N .v/ Df 2 L2 .t0 ; TI W/ W .x; t/ 2 g.x; v.x; t//

a.e. on

C .t0 ; T/g:

If v; 2 L2 .t0 ; TI W/, then we have 2 N .v/

,

.t/ 2 N.v.t//

,

.x; t/ 2 g.x; v.x; t//

for a.e.

t 2 .t0 ; T/

for a.e.

.x; t/ 2 C .t0 ; T/:

Let 0 < 1 2 : : : k : : : be the sequence of the eigenvalues of the operator A, with the corresponding eigenvectors fzk g1 kD1 being orthonormal in H and orthogonal in V. Define the finite dimensional spaces Vk D span fz1 ; : : : ; zk g. S These spaces approximate V from the inside, i.e., 1 V is dense in V. By Pk v we kD1 k denote the projection onto Vk of the function v 2 H. Then jPk vj jvj and Pk v ! v strongly in H as k ! 1. We start with the following Galerkin problem. Problem 15.1. Find u 2 AC.Œt0 ; TI Vk / such that u.t0 / D Pk u0 and for all v 2 Vk and a.e. t 2 .t0 ; T/ we have Z

u0 v dxC

Z

˝

Z

Z

ruW rv dxC ˝

˝

.ur/uv dxC

C

v dS D hf ; vi;

(15.15)

with 2 N .u /. Theorem 15.3. Let .g1/–.g4/ hold. Problem 15.1 has at least one solution. Proof. We will prove the existence of solution to this ordinary differential inclusion by means of the Kakutani–Fan–Glicksberg fixed point theorem (see Theorem 3.8). Fix 2 L2 .t0 ; TI W/ and w 2 L2 .t0 ; TI V/, and consider the following auxiliary linear problem. Problem 15.2. Find u 2 AC.Œt0 ; TI Vk / such that u.t0 / D Pk u0 and for all v 2 Vk and a.e. t 2 .t0 ; T/ we have Z ˝

u0 .t/v dxC

Z

Z

Z

ru.t/W rv dxC ˝

˝

.w.t/r/u.t/v dxC

C

.t/v dS D hf ; vi: (15.16)

This linear system of ODEs has a unique solution u. We define a multifunction 2 W V L2 .t0 ; TI W/ ! V 2L .t0 ;TIW/ which assigns to the pair .w; / the set fug N .u /. First, we note that by assertion .H1/ of Theorem 3.23, the set N .u / is nonempty and convex, so the multifunction assumes nonempty and convex values. We will prove, by the Kakutani–Fan–Glicksberg fixed point theorem (see Theorem 3.8), that this multifunction has a fixed point. This fixed point is a solution of Problem 15.1. Taking v D u.t/ in (15.16), we get 1d ju.t/j2 C ku.t/k2 k.t/kW k kku.t/k C kf kV 0 ku.t/k: 2 dt

346


It follows that 2kf k2V 0 d 2k k2 ju.t/j2 C ku.t/k2 k.t/k2W C : dt After integration on .t0 ; t/, for any t 2 .t0 ; T/ we get Z

2

ju.t/j C

t

ku.s/k2 ds ju0 j2 C

t0

2kf k2V 0 .t t0 / 2k k2 kk2L2 .t0 ;tIW/ C :

In particular, kuk2V

2kf k2V 0 .T t0 / 1 2k k2 2 ju0 j2 C kk C ; 2 L .t0 ;TIW/ 2 2

(15.17)

and ku.t/k2 Ck2 ju.t/j2 Ck2 ju0 j2 C

C2 2kf k2V 0 .T t0 / Ck2 2k k2 kk2L2 .t0 ;TIW/ C k ; (15.18)

where the constant Ck depends only on the dimension of the space Vk . Let 2 N .u/. We have Z k.t/kW D sup

kzkW D1 C

Z .x; t/ z.x/ dx sup

kzkW D1 C

.c1 C c2 ju.x; t/j/jz.x/j dx

p p c1 m2 .C / C c2 k u.t/kW c1 L1 L2 C c2 k kku.t/k: Hence k.t/k2W 2c21 L1 L2 C 2c22 k k2 ku.t/k2 2c21 L1 L2 C2c22 k k2

(15.19) ! 2 2 Ck2 2kf k2V 0 .Tt0 / 2 2 Ck 2k k 2 kkL2 .t0 ;TIW/ C Ck ju0 j C ;

and, after integration on .t0 ; T/, kk2L2 .t0 ;TIW/ 2.T t0 /.c21 L1 L2 C c22 k k2 Ck2 ju0 j2 / C

(15.20)

4c2 k k2 Ck2 kf k2V 0 .T t0 /2 4.T t0 /c22 Ck2 k k4 kk2L2 .t0 ;TIW/ C 2 :

Clearly, from (15.17) and (15.20), if only .T t0 / < and R2 > 0 such that is a map from S to 2 , where S

, 4c22 Ck2 kk4

we can find R1 > 0

S D f.u; / 2 V L2 .t0 ; TI W/ W kukV R1 ; kkL2 .0;TIW/ R2 g:


347

Consequently, the fixed point argument will give us the existence of solution on the interval Œt0 ; T for .T t0 / < 4c2 C2 kk4 , and by subsequent continuation this 2 k solution can be elongated to arbitrarily long time interval. It remains to prove that the graph of is weakly closed in S S. In view of the fact that V L2 .t0 ; TI W/ is a reflexive Banach space, and S is a bounded, closed, and convex set, it is enough to show the sequential weak closedness. Let wk ! w weakly in V and k ! weakly in L2 .0; TI W/, with .wk ; k / 2 S. Let moreover .uk ; k / 2 .wk ; k / be such that uk ! u weakly in V and k ! weakly in L2 .t0 ; TI W/. We know that .uk ; k / 2 S. Let us introduce an equivalent norm on the finite dimensional space Vk by the formula Z kzk

Vk0

D

sup v2Vk ;kvkD1 ˝

z v dx:

We have Z

ku0k .t/kVk0 D sup v2Vk

kvkD1

˝

u0k .t/ v dx D sup v2Vk

˝

˝

kvkD1

Z

Z hf ; vi ruk .t/W rv dx

Z

.wk .t/ r/uk .t/ v dx

C

k .t/ v dS

kf kV 0 C kuk .t/k C kwk .t/kkuk .t/k C kk .t/kW k k: Using (15.18) and (15.19) it follows from the fact that wk is uniformly bounded in L2 .t0 ; TI W/ that u0k is uniformly bounded in L2 .t0 ; TI Vk0 / and moreover in V , as the space Vk is finite dimensional. Hence, for a subsequence, not renumbered, u0k ! u0 weakly in V and uk ! u strongly in V . Moreover, .uk / ! u strongly in L2 .t0 ; TI W/. As k ! weakly in L2 .t0 ; TI W/ and k 2 N ..uk / /, by assertion .H2/ of Theorem 3.23 it follows that 2 N .u /. It remains to prove that u solves Problem 15.2 with w and , using the fact the uk solves this problem with wk and k . To this end, it suffices to multiply Eq. (15.16) written for uk ; wk ; k by the function 2 L2 .t0 ; T/ and integrate the resultant equation over the interval .t0 ; T/. The assertion follows by passing to the limit in the obtained equation. As the details are technical, their verification is left to the reader as an exercise. t u We are in position to define the Leray–Hopf weak solution of the considered problem. Since we need two types of weak solutions, we call this class a type I Leray–Hopf weak solution. Definition 15.5. By the type I Leray–Hopf weak solution of the three-dimensional Navier–Stokes problem with multivalued friction we mean a function 4

2 1 3 u 2 Lloc .Œt0 ; 1/I V/ \ Lloc .Œt0 ; 1/I H/ with u0 2 Lloc .Œt0 ; 1/I V 0 / such that:

• u.t0 / D u0 ; • for all v 2 V and a.e. t > t0 we have

348


hu0 .t/ C Au.t/ C B.u.t/; u.t// f ; vi C

Z C

.t/ v dS D 0;

(15.21)

2 with 2 Lloc .Œt0 ; 1/I W/ such that .t/ 2 N.u .t// for a.e. t > t0 , • for a.e. t t0 (including t0 ) and for all s > t we have

1 ju.s/j2 C 2

Z

s

Z

2

ku.r/k dr C

t

s

..r/; u .r//W hf ; u.r/i dr

t

1 ju.t/j2 : 2 (15.22)

Remark 15.3. Every type I Leray–Hopf weak solution defined above belongs to the spaces C.Œt0 ; 1/I V 0 / and Cw .Œt0 ; 1/I H/. Moreover, (15.22) implies that ju.t/j2 is continuous from the right at t0 . Theorem 15.4. Assume .g1/–.g5/. For any u0 2 H there exists at least one type I Leray–Hopf weak solution given by Definition 15.5. Proof. The proof is standard in the theory of three-dimensional Navier–Stokes equations and it is based on the Galerkin method. It is enough to prove, for certain given T > t0 , the existence of a function u 2 L2 .t0 ; TI V/ \ L1 .t0 ; TI H/ with 4 u0 2 L 3 .t0 ; TI V 0 / and corresponding 2 L2 .t0 ; TI W/ which satisfies the initial condition, (15.21) for a.e. t 2 .t0 ; T/, and (15.22) for a.e. t 2 .t0 ; T/ and for t D t0 . The solution on the interval .t0 ; 1/ can then be obtained by concatenating the weak solutions obtained on the intervals of length T t0 by taking the value of the previous solution at the endpoint of the time interval as the initial condition for the subsequent problem. We will proceed in the standard way obtaining the Leray–Hopf weak solution by passing to the limit in the Galerkin problems. Let uk 2 AC.Œt0 ; TI Vk / be the solutions of Problem 15.1 with the initial condition Pk u0 . We denote by k the corresponding selections of N ..uk / /. Taking v D uk .t/ in (15.15) we get for a.e. t 2 .t0 ; T/, 1d juk .t/j2 C kuk .t/k2 C 2 dt

Z C

k .t/ .uk / .t/ dS D hf ; uk .t/i:

(15.23)

Using .g4/ we get Z

Z C

jk .t/ .uk / .t/j dS

C

.c1 C c2 j.uk / .x; t/j/j.uk / .x; t/j dS

.c2 C 1/kuk .t/k2W C

c21 m2 .C / : 4

(15.24)

Let ı 2 .0; 12 / and let Z D V \ H 1ı .˝I R3 / be endowed with H 1ı .˝I R3 / topology. The embedding V Z is compact and Z H is continuous. Moreover, the trace operator from Z to W is linear and continuous. Thus, we can use the Ehrling lemma (cf. Lemma 3.17) to deduce that for any " > 0 there exists C."/ > 0 such that kvkW "kvk C C."/jvj for all v 2 V. Thus, from (15.24) we get


Z C

349

jk .t/ .uk / .t/j dS "kuk .t/k2 C C1 ."/juk .t/j2 C C2 ;

with constants C1 ."/; C2 > 0. Coming back to (15.23) we get 1d 1 juk .t/j2 C kuk .t/k2 2"kuk .t/k2 C kf kV 0 C C1 ."/juk .t/j2 C C2 : 2 dt 4"

(15.25)

It suffices to take " D 4 . Now, we use the Gronwall lemma (cf. Lemma 3.20) to deduce that uk is uniformly bounded in V and L1 .t0 ; TI H/. To prove that k is bounded in L2 .t0 ; TI W/ we observe that kk .t/k2W

Z C

.c1 C c2 j.uk / .x; t/j/2 dS 2c21 m2 .C / C 2c22 k k2 kuk .t/k2 ;

and the bound of uk in V implies that, indeed, k is bounded in L2 .t0 I TI W/. To obtain the estimates of u0k we will use the fact that we chose the eigenfunctions of the operator A asP a basis of the space Vk . The projection operator Pk W V 0 ! Vk is given by Pk v D kiD1 hv; zi izi , where zi is the i-th eigenfunction of the operator A. Define W V ! W by the tangential component of the trace, i.e., v D . v/ and W W ! V 0 its adjoint operator given by h v; wi D .v; w /W . Then we have Z C

k .t/ v dS D h k .t/; vi:

The Galerkin equation (15.15) can be rewritten in the following way: u0k .t/ C Auk .t/ C Pk B.uk .t/; uk .t// C Pk k .t/ D Pk f : As, by Theorem 3.17, we have kPk vkV 0 kvkV 0 , it follows that ku0k .t/kV 0 kAkL .VIV 0 / kuk .t/k C kf kV 0

(15.26)

C k kL .WIV 0 / kk .t/kW C kB.uk .t/; uk .t//kV 0 : Using (15.13) and (15.14) we get 1

3

kB.uk .t/; uk .t//kV 0 Cjuk .t/j 2 kuk .t/k 2 : 4

The last two bounds imply that u0k is bounded in L 3 .t0 ; TI V 0 /. Thus, for a subsequence, not renumbered, we have uk ! u weakly in V ;

(15.27) 1

uk ! u weakly- in L .t0 ; TI H/;

(15.28)

350


u0k ! u0

4

weakly in L 3 .t0 ; TI H/;

(15.29)

L2 .t0 ; TI H/;

(15.30)

uk ! u strongly in uk .t/ ! u.t/

strongly in

C.Œt0 ; TI V 0 /;

uk ! u strongly in uk .t/ ! u.t/ k ! .uk / ! u

for almost all t 2 .t0 ; T/;

H

weakly in H

(15.32)

t 2 Œt0 ; T;

for all

(15.33)

weakly in L2 .t0 ; TI W/;

(15.34)

L2 .t0 ; TI W/:

strongly in

(15.31)

(15.35)

The convergence (15.30) follows from the compact embedding 4

fu 2 V W u0 2 L 3 .t0 ; TI V 0 /g

L2 .t0 ; TI H/;

the convergence (15.32) follows from the compact embedding 4

fu 2 L1 .t0 ; TI H/ W u0 2 L 3 .t0 ; TI V 0 /g

C.Œt0 ; TI V 0 /;

L2 .0; TI Z/;

while (15.34) follows from the compact embedding 4

fu 2 L2 .t0 ; TI V/ W u0 2 L 3 .t0 ; TI V 0 /g

and the continuity of the Nemytskii trace operator from L2 .0; TI Z/ to L2 .0; TI W/. The convergence (15.33) follows by the computation Z .uk .t/; v/ D .Pk u0 ; v/ C

t t0

hu0k .s/; vi ds;

valid for v 2 H, the strong convergence Pk u0 ! u0 in H, and (15.29). Multiplying (15.15) by ' 2 C1 ..t0 ; T// and integrating over .t0 ; T/ we have, for v 2 Vk , Z

T

t0

hu0k .t/ C Auk .t/ C B.uk .t/; uk .t// f ; v'.t/i dt C

Z

T

.k .t/; v '.t//W dt D 0:

t0

Using (15.27), (15.29), (15.30), (15.33), and Lemma 15.1 we can pass to the limit in above equation, getting Z

T t0

hu0 .t/ C Au.t/ C B.u.t/; u.t// f ; v'.t/i dt C

Z

T

t0

..t/; v '.t//W dt D 0;

15.4 Existence and Invariance of Weak Global Attractor, and Weak Tracking. . .

351

S1 1 valid for v 2 kD1 Vk and ' 2 C ..t0 ; T//, whence (15.21) follows. Integrating (15.23) on .t; s/ we get, for all t; s with t0 t < s T, 1 juk .s/j2 C 2

Z

s

kuk .r/k2 dr C

Z

t

s

.k .r/; .uk / .r//W hf ; uk .r/i dr

t

1 juk .t/j2 : 2 (15.36)

Using (15.27), (15.31), (15.34), and (15.35) as well as the sequential weak lower semicontinuity of the norm of the space L2 .t; sI V/ we get, for a.e. t 2 .t0 ; T/ and a.e. s 2 .t; T/, Z

1 ju.s/j2 C 2

s

Z

ku.r/k2 dr C

t

s


t

1 ju.t/j2 : 2

The convergence (15.33) and the sequential weak lower semicontinuity of the norm j j imply that the last inequality holds for all s 2 Œt; T. Integrating (15.23) on .t0 ; s/ we get, for all s 2 .t0 ; T, 1 juk .s/j2 C 2

Z

s

2

Z

kuk .r/k dr C

t0

s

.k .r/; .uk / .r//W hf ; uk .r/i dr

t0

1 jPk u0 j2 : 2

As we did in (15.36), we can pass to the limit in the above inequality, to get 1 ju.s/j2 C 2

Z

s

t0

ku.r/k2 dr C

Z

s

t0


1 ju0 j2 : 2

We have proved (15.22). The fact that u.t0 / D u0 follows from (15.33) for t D t0 and the fact that uk .0/ D Pk u0 ! u0 strongly in H as k ! 1. Finally, (15.34), (15.35), and assertion .H2/ of Theorem 3.23 imply that 2 N .u /. The proof is complete. t u Remark 15.4. There are two “sources” of possible nonuniqueness of Leray–Hopf weak solution of the studied problem. First, the nonlinear term B which, in three-dimensional case, makes the weak solution uniqueness unknown (cf., e.g., [99, 219]), and, moreover, the nonmonotone friction multifunction g which also can contribute to the lack of solution uniqueness.

15.4 Existence and Invariance of Weak Global Attractor, and Weak Tracking Property First we discuss the dissipativity of the type I Leray–Hopf weak solutions. We recall a useful lemma.

352


Lemma 15.2 (cf. [11, Lemma 7.2]). Let W Œt0 ; 1/ ! R be lower semicontinuous, continuous at t0 , 2 L1 .0; T/ for all T > 0 and let, for some constant c 0 Z s

./ d .t/;

.s/ C c t

for all s t, for a.e. t > t0 and for t D t0 . Then

.t/ .t0 /ec.tt0 /

for all

t t0 :

We use the above lemma to get a dissipativity result for type I Leray–Hopf weak solutions. Lemma 15.3. If u is a type I Leray–Hopf weak solution given by Definition 15.5, then for all t t0 we have ju.t/j2 e1 .d2 kk

2 /.tt / 0

ju0 j2 C

kf k2V 0 C 2d1 L1 L2 . d2 k k2 / : 1 . d2 k k2 /2

(15.37)

Proof. Using .g5/ in (15.22) we get 1 ju.s/j2 C 2

Z

s

2

ku.r/k dr C

t

1 ju.t/j2 C " 2

Z sZ t

Z

s

C

d1 d2 ju .x; r/j2 dS dr

ku.r/k2 dr C

t

kf k2V 0 .s t/ ; 4"

for all s t and for a.e. t > t0 and for t D t0 , where " > 0 is arbitrary. Using the trace inequality, we get 1 ju.s/j2 C. d2 k k2 "/ 2 We take " D we get

d2 kk2 . 2

Z

s t

! kf k2V 0 1 2 Cd1 L1 L2 .st/: ku.r/k dr ju.t/j C 2 4" 2

Using the Poincaré inequality 1 jvj2 kvk2 valid for v 2 V,

ju.s/j2 C 1 . d2 k k2 /

Z

s

ju.r/j2 dr

t

! kf k2V 0 C 2d1 L1 L2 .s t/: d2 k k2

ju.t/j2 C

Denote A D 1 . d2 k k2 / and B D ju.s/j2

B CA A

Z

s t

kf k2V 0 d2 kk2

ju.r/j2

C 2d1 L1 L2 . We have B B dr ju.t/j2 : A A


353

As u 2 Cw .Œt0 ; 1/I H/ and ju.t/j2 is continuous from the right at t0 (cf. Remark 15.3), it follows that the function .r/ D ju.r/j2 BA satisfies all assumptions of Lemma 15.2. It follows that B B A.tt0 / ju0 j2 e ju.t/j2 for all t t0 ; A A t u

whence we get the assertion of the lemma. The proof is complete. Define X D fu 2 H W juj2 Rg; kf k2V 0 C2d1 L1 L2 .d2 kk2 / . 1 .d2 kk2 /2

where R is any number strictly greater than from (15.37) we get the following result.

Directly

Lemma 15.4. Let K H be a bounded set, and let t0 2 R. For every type I Leray– Hopf weak solution u with the initial condition u.t0 / D u0 2 K there exists a time t1 D t1 .t0 ; K/ such that for all t t1 u.t/ 2 X. We endow X with two metrics. The strong metric ds W X X ! Œ0; 1/ is given by ds .u; v/ D ju vj. As the weak topology of H is metrizable on the closed ball X we can define the metric dw W X X ! Œ0; 1/ such that dw .uk ; u/ ! 0 if and only if uk ! u weakly in H. Note that the set X is compact in dw . We need to define a class of type II Leray–Hopf weak solutions (see [65, 66]). Definition 15.6. By the type II Leray–Hopf solution of the three-dimensional Navier–Stokes problem with multivalued friction we mean a function u 2 4

2 1 3 Lloc .Œt0 ; 1/I V/ \ Lloc .Œt0 ; 1/I H/ with u0 2 Lloc .Œt0 ; 1/I V 0 / such that:

• for all v 2 V and a.e. t > t0 we have hu0 .t/ C Au.t/ C B.u.t/; u.t// f ; vi C

Z C

.t/ v dS D 0;

(15.38)

2 with 2 Lloc .t0 ; 1I W/ such that .t/ 2 N.u .t// for a.e. t > t0 , • for a.e. t t0 (not necessarily for t0 ) and for all s > t we have

1 ju.s/j2 C 2

Z t

s

ku.r/k2 dr C

Z t

s


1 ju.t/j2 : 2 (15.39)

Note that every type I Leray–Hopf weak solution is a type II Leray–Hopf weak solution. Moreover, a restriction of either type I or type II Leray–Hopf weak solution given on interval Œt0 ; 1/ to a smaller interval ŒT; 1/ Œt0 ; 1/ is always a type II Leray–Hopf weak solution on ŒT; 1/. It is, however, impossible to concatenate

354


type II Leray–Hopf weak solutions, as the energy inequality (15.39) does not have to hold at the initial time t D t0 . We can define E .ŒT; 1// Dfu W u

is a type II Leray–Hopf weak solution on

and

u.t/ 2 X

for all

ŒT; 1/;

t Tg:

(15.40)

Obviously, E .ŒT; 1// satisfies the first three axioms of Definition 15.1, and hence it defines the evolutionary system with E ..1; 1// given by axiom .iv/ of Definition 15.1. We can use Theorem 15.1 to conclude the following result on the existence of a weak global attractor. Theorem 15.5. The evolutionary system defined by (15.40) has a weak global attractor A . To study the attractor invariance we need the next result. Theorem 15.6. Let uk be the sequence of type II Leray–Hopf weak solutions on Œt0 ; 1/ such that uk .t/ 2 X for all t t0 . Let T > t0 be given arbitrarily. Then there exists a subsequence, not renumbered, which converges in Cw .Œt0 ; TI H/ to some type II Leray–Hopf weak solution u, i.e., .uk .t/; v/ ! .u.t/; v/

uniformly on

Œt0 ; T

for all v 2 H

as

k ! 1:

Proof. Let the sequence tm ! t0 be such that (15.39) holds for t D tm . Proceeding in the same way as in the proof of Lemma 15.3 we get Z

T

kuk .r/k2 dr C;

tm

from (15.39). Passing with m ! 1 we obtain that uk is bounded in L2 .t0 I TI V/. Proceeding as in (15.26) we find that from (15.38) it follows that u0k is bounded 4 in L 3 .t0 ; TI V 0 /. In the similar way as we obtained (15.27)–(15.35) in the proof of existence, we have uk ! u

weakly in V ;

uk ! u

weakly- in L1 .t0 ; TI H/;

u0k ! u0

weakly in L 3 .t0 ; TI H/;

uk ! u

strongly in L2 .t0 ; TI H/;

uk .t/ ! u.t/ uk ! u uk .t/ ! u.t/

4

strongly in H

for almost all t 2 .t0 ; T/;

strongly in C.Œt0 ; TI V 0 /; weakly in H

for all t 2 Œt0 ; T;

(15.41)


k ! .uk / ! u

355

weakly in L2 .t0 ; TI W/; strongly in L2 .t0 ; TI W/:

Now, exactly the same as in the proof of Theorem 15.4, it follows that u satisfies (15.38) and (15.39) for a.e. t 2 .t0 ; T/, whence u is a certain type II Leray–Hopf weak solution. To show the uniform convergence in Cw .Œt0 ; TI H/, let v 2 H and let " > 0. We can find v" 2 V with jv" vj ". We have sup j.uk .t/ u.t/; v/j D sup jhuk .t/ u.t/; v" i C .uk .t/ u.t/; v v" /j

t2Œt0 ;T

t2Œt0 ;T

p sup kuk .t/ u.t/kV 0 kv" k C 2 R"; t2Œt0 ;T

where we have used the fact that uk .t/; u.t/ 2 X. We pass with k ! 1 and use (15.41), to get p lim sup sup j.uk .t/ u.t/; v/j 2 R": k!1 t2Œt0 ;T

As " was arbitrary, it follows that limk!1 supt2Œt0 ;T j.uk .t/ u.t/; v/j D 0, and the proof is complete. t u Theorem 15.7. E .Œ0; 1// is compact in Cw .Œ0; 1/I H/. Proof. The proof follows the lines of the proof of Lemma 3.12 in [66]. Take any sequence uk 2 E .Œ0; 1//. By Theorem 15.6 there exists a subsequence, still denoted by uk , which converges in Cw .Œ0; 1I H/ to some u1 , a type II Leray–Hopf solution. Passing again to subsequence, not renumbered, uk converges in Cw .Œ0; 2I H/ to some u2 , a type II Leray–Hopf solution such that u1 D u2 on Œ0; 1. Continuing this diagonalization process, uk converges in Cw .Œ0; 1/I H/ to some u, a type II Leray–Hopf solution. As ju.t/j lim infk!1 juk .t/j for all t 0, it follows that u.t/ 2 X, and in consequence u 2 E .Œ0; 1//. The proof is complete. t u As we have shown that .A1/ holds, we can use Theorem 15.2 to conclude the last result of this section. Theorem 15.8. The weak global attractor A is a maximal invariant set. Moreover, A D fx 2 X W x D u.0/

for some

u 2 E ..1; 1//g;

and the tracking property holds, i.e., for every " > 0 there exists t0 > 0 such that for any t > t0 and for any trajectory u 2 E .Œ0; 1// we have dCw .Œt ;1/IX/ .u; v/ < " for a certain complete trajectory v 2 E ..1; 1//.

356


The tracking property established in Theorem15.8 is illustrated in Fig. 15.3. Fig. 15.3 Illustration of the tracking property. Every trajectory has a corresponding complete trajectory on the attractor to which it converges as the time goes to infinity

15.5 Comments and Bibliographical Notes There are several mathematical formalisms useful to study the global attractors for autonomous evolution problems without uniqueness of solutions. Among them we can list: • generalized semiflows introduced and studied by Ball [11], • multivalued semiflows (m-semiflows) anticipated by Babin and Vishik [9] and introduced and studied by Melnik and Valero in [171] and [172], • trajectory attractors introduced independently by Chepyzhov and Vishik [59], Málek and Neˇcas [167], and Sell [210]. The application of this approach to twodimensional incompressible Navier–Stokes problem with multivalued feedback control boundary condition is presented in Chap. 14, • evolutionary systems introduced by Cheskidov and Foia¸s and [66] and later extended by Cheskidov in [65]. The formalism of evolutionary systems chosen by us in this chapter appears to be especially well suited to the problems governed by the three-dimensional Navier– Stokes equations. Still, such problems can be and have been dealt with by the other methods: trajectory attractors [59, 167, 210], multivalued semiflows [128, 141], and generalized semiflows [11]. The approach based on generalized semiflows has been compared with the one based on multivalued semiflows in [40], and both these approached were related to the one based on the trajectory attractors in [129]. More information on the structure of weak attractor for three-dimensional case with periodic boundary conditions can be found in [102] and on the structure of the attainability sets of weak solutions in [138]. If we assume the periodic or homogeneous Dirichlet boundary conditions on whole @˝ it is known that a weak solution for three-dimensional Navier–Stokes equations, so-called Leray–Hopf solution (see [115, 154]), exists for arbitrarily long time but its uniqueness is still an open problem. On the other hand, the regular solution is known to exist and be unique only locally in time (see [98] for local


357

regularity results in three-dimensional case and global regularity results in twodimensional case for periodic boundary conditions). There are many additional conditions under which the weak solution becomes strong and the associated weak global attractor is the strong one, too, but so far it remains an open problem whether any of such conditions is in fact satisfied [34, 66–68, 73, 82, 143, 144, 189]. Only such conditional results are available on the existence of strong global attractor in the three-dimensional case [65, 66, 128, 202]. An interesting recent idea is to use the results of numerical simulations to verify the existence of strong solutions [64]. A lot of work has been put in recent years to study the partial regularity of solutions. This work includes the analysis of the so-called singular set of the weak solutions, i.e., the set of space-time points such that the solution is unbounded in the neighborhood of such points, this work started from the works of Scheffer [205] and Caffarelli et al. [32]. For the recent progress in such studies see the forthcoming book [200]. There are some approaches relying on the modifications or simplifications of three-dimensional equations to improve their properties, for example, one can study globally modified Navier–Stokes equations [47, 48, 139], Navier–Stokes equations with fractional operators [86], or ˛-model [45, 63, 100].

16

Attractors for Multivalued Processes in Contact Problems

In this chapter we consider further non-autonomous and multivalued evolution problems, this time in the frame of the theory of pullback attractors for multivalued processes. First we prove an abstract theorem on the existence of a pullback D-attractor and then apply it to study a two-dimensional incompressible Navier–Stokes flow with a general form of multivalued frictional contact conditions. Such conditions represent the frictional contact between the fluid and the wall, where the friction force depends in a nonmonotone and even discontinuous way on the slip rate, and are a generalization of the conditions considered in Chap. 10. In contrast to Chap. 10 we are able to obtain only the attractor existence, while, for example, the question of its fractal dimension for the case without uniqueness remains an open problem.

16.1 Abstract Theory of Pullback D-Attractors for Multivalued Processes In this section we recall basic definitions of the theory of pullback D-attractors for multivalued processes, prove a theorem which gives some useful criteria of existence of pullback D-attractors for such processes, and introduce the notion of “condition (NW).” Let .H; %/ be a complete metric space, and P.H/ be the family of all nonempty subsets of H. By distH .A; B/ we denote the Hausdorff semi-distance between the sets A; B H defined as distH .A; B/ D sup inf %.x; y/: x2A y2B


359

360

16 Attractors for Multivalued Processes in Contact Problems

If the set A H is bounded, than we define its Kuratowski measure of noncompactness .A/ (cf. [144]) as

.A/ D finf ı > 0 W A has finite open cover of sets of diameter less than ı g: For the properties of , see, for example, [111, 125]. Subsets of R H will be called non-autonomous sets. For a non-autonomous set D R H we identify D with the family of its fibers, i.e., D D fD.t/gt2R ;

where

x 2 D.t/ , .x; t/ 2 D:

A family D of non-autonomous sets such that for every D 2 D and all t 2 R the fiber D.t/ is nonempty (i.e., D.t/ 2 P.H/) is called an attraction universe over H. We denote Rd D f.t; / 2 R2 W t g. Definition 16.1. The map U W Rd H ! P.H/ is called a multivalued process (m-process) if: (1) U.; ; z/ D z for all z 2 H, (2) U.t; ; z/ U.t; s; U.s; ; z// for all z 2 H and all t s . If in (2), in place of inclusion we have the equality U.t; ; z/ D U.t; s; U.s; ; z//, then the m-process is said to be strict. Definition 16.2. Let D be an attraction universe over H. The m-process U in H is pullback !-D-limit compact if for every D D fD. /g 2R 2 D and t 2 R we have

[

! U.t; ; D. // ! 0

as

s ! 1:

s

Definition 16.3. Let D be an attraction universe over H. The m-process U in H is pullback D-asymptotically compact if for any t 2 R, every D 2 D, and all sequences k ! 1 and k 2 U.t; k ; D.k //, there exists a subsequence fkm g such that for some 2 H, km ! in H. Lemma 16.1. Let D be an attraction universe over H. The m-process U in H is pullback !-D-limit compact if and only if it is pullback D-asymptotically compact. Proof. We first prove that pullback !-D-limit compactness implies pullback D-asymptotic compactness. Fix t 2 R. Let D 2 D and let k be such that ! [ 1 and k ! 1: U.t; ; D. // (16.1)

k k

Let ti ! 1 and i 2 U.t; ti ; D.ti //. We shall prove that .fi g1 iD1 / D 0. For every m 2 N we have m 1 1

.fi g1 iD1 / D .fi giD1 [ fi giDmC1 / .fi giDmC1 /:

16.1 Abstract Theory of Pullback D-Attractors for Multivalued Processes

361

1 For every k 2 N and m such that tmC1 k we have .fi g1 iDmC1 / k , 1 1 whence .fi giD1 / D 0. This proves, in turn, the precompactness of fi giD1 , and, in consequence, the pullback D-asymptotic compactness of U. Now let U be pullback D-asymptotically compact. We shall prove first that for every D 2 D and t 2 R the set

!.D; t/ D

\[

U.t; ; D. //

(16.2)

st s

is nonempty. Indeed, let tk ! 1 and k 2 U.t; tk ; D.tk // and let, for a subsequence, still denotedSby k, k ! . For every s t and every indexSk such that tk s, we have k 2 s U.t; ; D. //. Since k ! , then 2 s U.t; ; D. // for every s t. Thus 2 !.D; t/. Now we prove that for every D 2 D and every t 2 R, distH .U.t; ; D. //; !.D; t// ! 0 as

! 1:

Assume, to the contrary, that there exists the non-autonomous set D 2 D, the number Nt 2 R, and the sequences tk ! 1 and k 2 U.Nt; tk ; D.tk // such that distH .k ; !.D; t// > 0. By the asymptotic compactness property, for a subsequence, still denoted by k, we have k ! in H. But 2 !.D; t/, which gives a contradiction. Let D 2 D, t 2 R and let xn be a sequence in !.D; t/. We prove that this sequence has a subsequence that converges to some element in !.D; t/ and thus the set !.D; t/ is compact. As xn 2

[

U.t; ; D. //

for every

s t;

s

then, for any sequence tk ! 1 there exists mk 2 U.t; tmk ; D.tmk // such that .xk ; mk / 1k . By pullback D-asymptotic compactness, there exists a subsequence of mk converging to some 2 !.D; t/. Thus, also x ! . S Now let us fix > 0. We need to show that there exists s < t such that the set s U.t; ; D. // can be covered by a finite number of sets with diameter . From compactness of !.D; t/ it follows that there exists a finite number of points fxi gkiD1 S such that !.D; t/ kiD1 B.xi ; 2 /: Now choose s < t such that distH .U.t; ; D. //; !.D; t// <

" ; 2

whenever s. It follows that for all s we have G.t; ; D. // and the proof is complete.

Sk iD1

B.xi ; / t u

362


Definition 16.4. Let D be an attraction universe over H. We say that an m-process U has a pullback D-absorbing non-autonomous set B D fB.t/gt2R 2 D if for every t 2 R and D 2 D there exists 0 t depending on t and D such that for 0 , U.t; ; D. // B.t/: Definition 16.5. Let D be an attraction universe over H and let U be an m-process on H. The non-autonomous set A D fA.t/gt2R R H is called a pullback Dattractor for U if: (1) A.t/ is compact in H and nonempty for every t 2 R, (2) A.t/ U.t; ; A. // for every t 2 R and t (A is negatively semi-invariant), (3) A is pullback D-attracting, that is, for every t 2 R and D 2 D, distH .U.t; ; D. //; A.t// ! 0 as

! 1;

(4) if C is a non-autonomous set such that C.t/ are closed for t 2 R and C is pullback D-attracting then A.t/ C.t/ for every t 2 R (minimality property). Remark 16.1. Condition (4) in the above definition implies that pullback D-attractor is always defined uniquely. Remark 16.2. If we assume that, by definition, A 2 D then condition (4) (minimality) follows from conditions (1)–(3). In such a case, by Remark 16.1, conditions (1)–(3) suffice to guarantee the uniqueness of A. The minimality property can be proved as follows. Suppose that there exists a non-autonomous set C such that C.t/ are closed for all t 2 R, C is attracting, and for some t 2 R, C.t/ A.t/ with proper inclusion. As A 2 D, we have distH .U.t; ; A. //; C.t// ! 0 for ! 1. But A.t/ U.t; ; A. //, and we have distH .A.t/; C.t// distH .U.t; ; A.t//; C.t// ! 0 as ! 1, whence A.t/ C.t/, a contradiction. Definition 16.6. The m-process U on H is closed if for all t the graph of the multivalued mapping x ! U.t; ; x/ is a closed set in the product topology on HH. Now we formulate a theorem about the existence of pullback D-attractors in complete metric spaces. A similar result is proved in [4, 35, 169] (see also [171, 172] for the case of m-semiflows and [41, 239] for the case of the universe of bounded and constant in time sets). Theorem 16.1. Let H be a complete metric space and let the m-process U on H be closed. Assume that (i) U has a pullback D-absorbing non-autonomous set B 2 D. (ii) U is pullback !-D-limit compact (equivalently, U is pullback D-asymptotically compact). Then there exists a pullback D-attractor for U.


363

Proof. The proof is similar to that of Theorem 2.1 in [125]. Step 1.

We define a candidate set for a pullback D-attractor by setting A.t/ D

\[

U.t; ; B.//;

(16.3)

st s

where B D fB. /gt2R is a non-autonomous pullback D-absorbing set. Step 2. A.t/ is nonempty and compact. We have, by !-D-limit compactness of U,

[

! U.t; ; B. // ! 0

as

s ! 1:

(16.4)

s

S Since the sets Xs D s U.t; ; B. // are nonempty, bounded, and closed in H, and the family of sets fXs gst is nonincreasing as s ! 1, we can apply a well-known property of (cf., e.g., [111]) to get the claims. Step 3. Attraction property. Since for every t 2 R, every non-autonomous set D 2 D, and every r t it is U.t; ; D. // U.t; r; U.r; ; D. /// U.t; r; B.r// for all 0 , for some 0 .D; r/ r, it suffices to prove that distH .U.t; ; B. //; A.t// ! 0 as ! 1. We argue by contradiction. Assume, to the contrary, that there exist sequences k ! 1 and k 2 U.t; k ; B.k // such that infy2A.t/ %.k ; y/ > 0. By the pullback D-asymptotic compactness property, there exists a convergent subsequence, ! in H, and by the definition of the pullback D-attractor in (16.3), 2 A.t/, which gives a contradiction. Step 4. We prove that A.t/ U.t; ; A. // for all t. Let x 2 A.t/ and t. We shall prove that x 2 U.t; ; p/ for some p 2 A. /. Since x 2 A.t/ defined in (16.3), there exist tk ! 1 and k 2 U.t; tk ; B.tk // such that k ! x in H. We have k 2 U.t; tk ; B.tk // U.t; ; U.; tk ; B.tk /// for large k, whence there exist zk 2 B.tk / such that k 2 U.t; ; U.; tk ; zk //, and pk 2 U.; tk ; zk / such that k 2 U.t; ; pk /. By the pullback D-asymptotic compactness it follows that for a subsequence of fpk g we have p ! p in H. From (16.3) it follows that p 2 A. /. Since ! x and p ! p with 2 U.t; ; p /, from the fact that U is closed it follows that x 2 U.t; ; p/. The proof is complete. Step 5. To prove the minimality property assume that there exists a pullback Dattracting non-autonomous set C such that C.t/ are closed. Then due to the fact that B 2 D we have distH .U.t; ; B. //; C.t// ! 0 as ! 1 . Let x 2 A.t/. By (16.3) there exist sequences k ! 1 and k 2 U.t; k ; B.k // such that k ! x, whereas, as C.t/ is closed, we have x 2 C.t/ and the proof is complete. t u

364


Remark 16.3. The above theorem still holds if we assume that, in definition of the pullback D-asymptotic compactness, any set D 2 D is replaced with the pullback D-absorbing set B. In such a case this set does not have to belong to the universe D (see [4, 35, 169]). Remark 16.4. The existence of the pullback D-attractor implies condition (ii) in the above theorem, that is, the !-D-limit compactness of U. To prove it we follow the argument provided, e.g., in [240]. Denote by O" .A.t// the closed "-neighborhood of A.t/. Since the set A.t/ is compact, then .A.t// D 0, and .O" .A.t/// 2". From the attraction property of A, for any D 2 D we have distH .U.t; ; D. //; A.t// 2" for all 0 for some 0 .D; t/. Thus

[

! U.t; ; D. // 2";

0

which implies (ii). Remark 16.5. In general, the existence of a pullback D-absorbing non-autonomous set in D does not follow from the existence of a pullback D-attractor. Indeed, consider the equation u0 .t/ D 0 in R. Then U.t; / D S.t / D id. Let the universe D consist of only one non-autonomous set D, given by D. / D fsin g. Then the attractor is given as A.t/ D A D Œ1; 1 for t 2 R. There is no choice for a family of D-absorbing set but B. / D D. /. But U.t; ; D. // D D. / 6D B.t/ as sin 6D sin t. Thus, in this case there does not exist a pullback D-absorbing non-autonomous set in D. The presented example is illustrated in Fig. 16.1.

Fig. 16.1 Illustration of the example in Remark 16.5. The identity semiflow has a pullback D-attractor but it does not have a pullback D-absorbing non-autonomous set in D

It is natural to ask what assumptions on the universe D are required to guarantee that existence of pullback D-attractor implies (i). We make the following assumptions: (HD1) The universe D is inclusion-closed, that is if D D fD.t/g t2R 2 D and D0 D fD0 .t/gt2R is such that D0 .t/ D.t/ and D0 .t/ 2 P.H/ for all t 2 R, then D0 2 D. (HD2) If D D fD.t/gt2R 2 D, then there exists " > 0 such that the nonautonomous set fO" .D.t//gt2R also belongs to D. Corollary 16.1. Let the universe D satisfy (HD1) and let the m-process U satisfy the assumptions of Theorem 16.1. Then we have A 2 D. If we moreover assume that U is strict then A is invariant, i.e., U.t; ; A. // D A.t/ for all t 2 R and t.


365

Proof. The proof follows the lines of the proofs of item (6) in Theorem 3 in [169] and items (5) (6) in Theorem 3.10 in [4]. Since B is pullback D-absorbing we have distH .U.t; ; D. //; B.t// D 0 for all D 2 D and t0 .D; t/. Hence as ! 1 we have distH .U.t; ; D. //; B.t// ! 0 and fB.t/gt2R is pullback D-attracting. Then from the attractor minimality we have A.t/ B.t/ and the assertion follows from (HD1). For the proof of the positive invariance, let t be fixed and let r 0. We have U.t; ; A. // U.t; ; U.; C r; A. C r/// D U.t; C r; A. C r//: Since A 2 D, it pullback attracts itself, and lim distH .U.t; C r; A. C r//; A.t// D 0:

r!1

Hence lim distH .U.t; ; A. //; A.t// D 0

r!1

and distH .U.t; ; A. //; A.t// D 0, whence, by the closedness of A.t/, it follows that U.t; ; A. // A.t/ and the assertion is proved. t u Corollary 16.2. Let the universe D satisfy (HD2) and let the m-process U have the pullback D-attractor A 2 D. Then U has a pullback D-absorbing non-autonomous set B 2 D. Proof. Define B.t/ D O" .A.t//, where " is given in (HD2). Then B D fB.t/gt2R belongs to D. Suppose, on the contrary, that B.t/ is not pullback D-absorbing. This means that there exist sequences tk ! 1 and k 2 U.t; tk ; D.tk // such that for large k we have k 62 B.t/. From Remark 16.4 it follows that U is pullback D-asymptotically compact, and, for a subsequence, ! . As A is pullback D-attracting and A.t/ is closed, we have 2 A.t/, a contradiction with k 62 O" .A.t//. t u Summarizing, we have a following theorem which is a consequence of Theorem 16.1, Remark 16.4, and Corollaries 16.1 and 16.2. Theorem 16.2. Let D be a universe of non-autonomous sets satisfying (HD1) and (HD2) and let U be a closed m-process. Then U has a pullback D-attractor A belonging to D if and only if U is pullback D-asymptotically compact and has a pullback D-absorbing non-autonomous set B 2 D: In the sequel of this section we confine ourselves to Banach spaces where we will consider both strong and weak topologies. We introduce condition (NW), “norm-toweak,” that generalizes to the non-autonomous multivalued case the norm-to-weak continuity assumed in [240] for semigroups (see Definition 3.4 in [240]) and in [125] for multivalued semiflows (see Definition 2.11 in [125]). A similar condition is introduced for the non-autonomous multivalued case in [231] (see condition (3) in

366


Definition 2.6 in [231]), where only the strict case is considered and, instead of a subsequence, whole sequence is assumed to converge weakly. Definition 16.7. The m-process U on a Banach space H satisfies condition (NW) if for every t 2 R and t, from xk ! x in H and k 2 U.t; ; xk / it follows that there exists a subsequence fmk g, such that mk ! weakly in H with 2 U.t; ; x/. Lemma 16.2. If a multivalued process U on a Banach space H satisfies condition (NW) then it is closed. Proof. An elementary proof follows directly from the definitions.

t u

From Theorem 16.1 together with Lemma 16.2 we have the following Corollary 16.3. If the m-process U on a Banach space H has a pullback Dabsorbing non-autonomous set B 2 D, is pullback D-asymptotically compact (equivalently, !-D-limit compact), and satisfies condition (NW) then there exists a pullback D-attractor for U. The above corollary provides useful criteria of existence of pullback D-attractors for multivalued processes. In the next section we apply them to study the time asymptotic behavior of solutions of a problem originating from contact mechanics.

16.2 Application to a Contact Problem Problem Formulation The flow of an incompressible fluid in a two-dimensional domain ˝ is described by the equation of motion u0 .t/ u.t/ C .u.t/ r/u.t/ C rp.t/ D f .t/ in

˝ .t0 ; 1/;

(16.5)

and the incompressibility condition div u.t/ D 0 in ˝ .t0 ; 1/;

(16.6)

where the unknowns are the velocity u W ˝ .t0 ; 1/ ! R2 and pressure p W ˝ .t0 ; 1/ ! R, > 0 is the viscosity coefficient and f W ˝ .t0 ; 1/ ! R2 is the volume mass force density. To define the domain ˝ of the flow, let ˝1 be the infinite channel ˝1 D fx D .x1 ; x2 / 2 R2 W x2 2 .0; h.x1 // g; where h W R ! R is a positive function (we assume that h.x1 / h0 > 0 for all x1 2 R), smooth, and L-periodic in x1 . Then we set ˝ D fx D .x1 ; x2 / 2 R2 W x1 2 .0; L/; x2 2 .0; h.x1 // g;

16.2 Application to a Contact Problem

367

with the boundary @˝ D D [ C [ L , where D D f.x1 ; h.x1 // W x1 2 .0; L/g, C D .0; L/ f0g, and L D f0; Lg .0; h.0// are the top, bottom, and lateral parts of @˝, respectively. We will use the notation e1 D .1; 0/ and e2 D .0; 1/ for the canonical basis of R2 . Note that on C the outer normal unit vector is given by n D e2 . We are interested in the solutions of (16.5)–(16.6) periodic in the sense that for 2 ;t/ 2 ;t/ D @u2 .L;x , x2 2 Œ0; h.0/ and t 2 RC we have u.0; x2 ; t/ D u.L; x2 ; t/, @u2 .0;x @x1 @x1 and p.0; x2 ; t/ D p.L; x2 ; t/. The first condition represents the L-periodicity of velocities, while the latter two ones the L-periodicity of normal stresses in the space of divergence free functions. Moreover, we assume that u.t/ D 0 on

D .t0 ; 1/:

(16.7)

On the contact boundary C we decompose the velocity into the normal component un D u n, where n is the unit outward normal vector, and the tangential one u D u e1 . Note that since the domain ˝ is two-dimensional it is possible to consider the tangential components as scalars, for the sake of the ease of notation. Likewise, we decompose the stress on the boundary C into its normal component Tn D Tn n and the tangential one T D Tn e1 . The stress tensor is related to the velocity and pressure through the linear constitutive law Tij D pıij C .ui;j C uj;i /. We assume that there is no flux across C , un .t/ D 0

on C .t0 ; 1/;

(16.8)

and that the tangential component of the velocity u on C is in the following relation with the tangential stresses T , T .x; t/ 2 @j.x; t; u .x; t//

on C .t0 ; 1/:

(16.9)

In the above formula, j W C RR ! R is a potential which is locally Lipschitz and not necessarily convex with respect to the last variable, and @ is the subdifferential in the sense of Clarke (see Sect. 3.7) taken with respect to the last variable. Note that the function j is assumed to depend on both space and time variables, however, sometimes the dependence on the space variable x 2 C is skipped for the ease of notation. In the end, we have the initial condition u.x; t0 / D u0 .x/

in

˝:

(16.10)

Relation (16.9) is a generalized version of the Tresca friction law. We assume that the friction force is related by the multivalued and nonmonotone law with the tangential velocity. Our setup is more general than in the previous works [126, 162]. In contract to [126] we do not assume any monotonicity type conditions on the potential j and hence the solutions to the formulated problem can be nonunique here. Similar laws were used, for example, in [12, 213] for static problems in

368


elasticity, see Fig. 4.6 in [12] and Fig. 2(a) in [213] for examples of particular nonmonotone friction laws that satisfy the assumptions .j1/–.j4/ listed below. Complex nonmonotone behavior of the friction force density is related to the presence of asperities on the surface [191]. Similar friction law is also used, for example, in the study of the motion of tectonic plates, see [119, 190, 207] and the references therein. Weak Formulation and Existence of Solutions We introduce the variational formulation of the problem and, for the convenience of the readers, we describe the relations between the classical and the weak formulations. We begin with some basic definitions. Let VQ D fu 2 C1 .˝/2 W div u D 0 in ˝; u D 0 at D ;

u is L-periodic in x1 ; u n D 0 at C g



We define scalar products in H and V, respectively, by Z

Z .u ; w/ D

u.x/ w.x/ dx

and

˝

.ru ; rw/ D

ru.x/W rw.x/ dx; ˝


juj D .u; u/ 2

and

1

kuk D .ru ; ru/ 2 :

The duality in V 0 V is denoted as h; i. Moreover, let, for u; w, and z in V, a.u ; w/ D .ru ; rw/

and

b.u ; z ; w/ D ..u r/z ; w/:

We define a linear operator A W V ! V 0 as hAu; wi D a.u; w/, and a nonlinear operator hBu; wi D b.u; u; w/. For w 2 V we have the Poincaré inequality 1 jwj2 kwk2 , where 1 is the first eigenvalue of the Stokes operator A in V. The linear and continuous trace mapping leading from V to L2 .C /2 is denoted by , we will use the same symbol to denote the Nemytskii trace operator on the spaces of time-dependent functions. The norm of the trace operator will be denoted by k k k kL .VIL2 .C /2 / . 2 If u 2 L2 .C / we will denote by [email protected];u/ the set of all L2 selections of @j.t; u/, i.e., 2 all functions 2 L .C / such that .x/ 2 @j.x; t; u.x// for a.e. x 2 C (note that dependence on x 2 C is not written explicitly for the sake of notation brevity).


369

The variational formulation of the problem is as follows. Problem 16.1. Given u0 2 H, find u W .t0 ; 1/ ! H such that for all T > t0 , u 2 C.Œt0 ; TI H/ \ L2 .t0 ; TI V/

with u0 2 L2 .t0 ; TI V 0 /;

and hu0 .t/ C Au.t/ C Bu.t/; zi C ..t/; z /L2 .C / D hf .t/; zi;

(16.11)

2 .t/ 2 [email protected];u .t//

(16.12)

for a.e. t t0 and for all z 2 V. The assumptions on the problem data are the following. The functions f W R ! H and j W C R R ! R satisfy: .f 1/ .f 2/

2 f 2 Lloc .RI H/, for certain t 2 R (and thus for all t 2 R) we have

Z

t

1

e.dkk

2 / s 1

jf .s/j2 ds < 1;

.j1/ j.x; t; / is locally Lipschitz for a.e. .x; t/ 2 C R, .j2/ j.; ; s/ is measurable for all s 2 R, .j3/ @j satisfies the growth condition jj a C bjsj for all s 2 R and 2 @j.x; t; s/ with a; b > 0 a.e. .x; t/ 2 C R, .j4/ @j satisfies the s c djsj2 for all s 2 R and 2 @j.x; t; s/ with

condition c 2 R and d 2 0; kk2 a.e. .x; t/ 2 C R. 2 is in fact the set of all Note that by .j3/ it follows that for u 2 L2 .C / the set [email protected];u/ measurable selections of @j.t; u/. Condition .j4/ is known as dissipativity condition. We have the following relations between classical and weak formulations.

Proposition 16.1. Every classical solution of (16.5)–(16.10) is also a solution of Problem 16.1. On the other hand, every solution of Problem 16.1 which is smooth enough is also a classical solution of (16.5)–(16.10). Proof. The proof is only sketched here, the details are left to the reader. Let u be a classical solution of (16.5)–(16.10). As it is (by assumption) sufficiently regular, we have to check only (16.11). Remark first that (16.5) can be written as u0 .t/ div T.u.t/; p.t// C .u.t/ r/u.t/ D f .t/:

(16.13)

Let z 2 V. Multiplying (16.13) by z, integrating by parts and using the Green formula we obtain

370


Z ˝

u0 .t/ z dx C

Z Z

˝

Tij .u.t/; p.t//zi;j dx C b.u.t/; u.t/; z/

D @˝

Z Tij .u.t/; p.t//nj zi dS C

˝

f .t/ z dx

(16.14)

for t 2 .t0 ; 1/. As u.t/ and z are in V, after some calculations we obtain Z ˝

Tij .u.t/; p.t//zi;j dx D a.u.t/; z/:

(16.15)

Taking into account the boundary conditions we get Z

Z @˝

Tij .u.t/; p.t//nj zi dS D

C

Z T .u.t/; p.t//z dS C

Tn .u.t/; p.t//zn dS;

C

and the last integral equals zero as z satisfies (16.8) a.e. on C . Thus, (16.11) holds. Conversely, suppose that u is a sufficiently smooth solution to Problem 16.1. We have immediately (16.6)–(16.8) and (16.10). Now, let z be in the space .H01 .div; ˝//2 D fz 2 V W z D 0 on @˝ g: We take such z in (16.11) to get hu0 .t/ u.t/ C .u.t/ r/u.t/ f .t/ ; zi D 0 for every

z 2 .H01 .div; ˝//2 :

Thus, there exists a distribution p.t/ on ˝ such that u0 .t/ u.t/ C .u.t/ r/u.t/ f .t/ D rp.t/

˝ .t0 ; 1/; (16.16)

in

so that (16.5) holds. Now, we shall derive the subdifferential boundary condition (16.9) from the weak formulation. Take z 2 V. We have Z Z Z Tij .u.t/; p.t//zi;j dx D div T.u.t/; p.t// z dx C Tn z dS: (16.17) ˝

˝

@˝

Since z 2 V, we have zn D 0 a.e. on C and hence Tn z D T z on C . Applying (16.15) and (16.17) to (16.11) we get Z ˝

.u0 .t/ div T.u.t/; p.t// C .u.t/ r/u.t/ f .t// z dx Z

Z C L

T.u.t/; p.t//n z dS C ..t/; z /L2 .C / C

C

T .t/z dS D 0:


371

By (16.16) we have (16.13) and so the first integral on the left-hand side vanishes. In the same way as in the proof of Proposition 10.1 we assume enough smoothness of the classical solution, such that the Cauchy stress vector Tn on L is L-periodic with respect to x1 . This means that the second integral R on the left-hand side of the last equality vanishes for any z 2 V. Thus we get C .T /z dS. Analogously to the proof of Proposition 10.1, we use the results of Chapter 1.2 in [146] on the extension of smooth functions on the boundary to divergence free vector fields to deduce that in the last inequality z can be replaced by any sufficiently smooth function. It follows that T D for a.e. x 2 C and the proof is complete. t u Theorem 16.3. Assuming .f 1/, and .j1/–.j3/, Problem 16.1 has a solution. The proof uses the standard approach by the mollification of the nonsmooth term and the Galerkin method. Since it is, on one hand, technical and quite involved, and, on the other hand the methodology is the same as in the proof of Theorem 10.8 in Chap. 10 and Theorem 14.1 in Chap. 14, with the necessary modifications for the non-autonomous terms, we skip the proof here. Multivalued Process and Its Attractor We associate with Problem 16.1 the multifunction U W Rd H ! P.H/, where U.t; t0 ; u0 / is the set of states attainable at time t from the initial condition u0 taken at time t0 . Observe that U is a strict m-process. 2 Lemma 16.3. Assume .f 1/, .j1/–.j4/. If the function u 2 Lloc .t0 ; C1I V/ with 2 0 0 u 2 Lloc .t0 ; C1I V / solves Problem 16.1, then

d ju.t/j2 C . dk k2 /ku.t/k2 C1 .1 C jf .t/j2 /; dt

(16.18)

ku0 .t/kV 0 C2 .1 C jf .t/j/ C C3 .1 C ju.t/j/ku.t/k

(16.19)

for a.e. t 2 .t0 ; C1/ with C1 ; C2 ; C3 > 0 independent on t0 ; u0 ; t. Proof. We take z D u.t/ in (16.11) and make use of b.w; w; w/ D 0

for

w2V

(16.20)

to get 1d ju.t/j2 C ku.t/k2 C ..t/; u .t//L2 .C / D .f .t/; u.t//; 2 dt 2 for a.e. t t0 . From .j4/ we obtain, with arbitrary " > 0, where .t/ 2 [email protected];u .t//

1d " 1 ju.t/j2 C ku.t/k2 ku.t/k2 C kf .t/k2V 0 C dku .t/k2L2 .C / cL: 2 dt 2 2"

372


Taking " D dk k2 we obtain, with a constant C > 0, 1d dk k2 ju.t/j2 C ku.t/k2 C.1 C kf .t/k2V 0 /; 2 dt 2 which proves (16.18), as H V 0 is a continuous embedding. To show (16.19) let us observe that for z 2 V and a.e. t 2 RC we have hu0 .t/; zi kAkL .VIV / ku.t/kkzkCkf .t/kV 0 kzk Ck.t/kL2 .C / k kkzk C cju.t/jku.t/kkzk; 2 , where we used the fact that, in view of the Ladyzhenskaya with .t/ 2 [email protected] .t// inequality (cf. Lemma 10.2), for w; z 2 V we have

jb.w; w; z/j cjwjkwkkzk:

(16.21)

Now (16.19) follows directly from the growth condition .j3/ and the trace inequality. t u Denote D . dk k2 /1 and define R.t/ by R.t/2 D

C1 C C1

Z

t

1

es jf .s/j2 ds C 1

for t 2 R:

(16.22)

By .f 2/ the quantity R.t/ is well defined and finite. We denote by R the family of all functions r W R ! .0; 1/ such that lim et r2 .t/ D 0:

t!1

By D we denote the family of all non-autonomous sets D D fD.t/gt2R such that D.t/ 2 P.H/ for all t 2 R and there exists rD 2 R such that for all t 2 R and for all w 2 D.t/, jwj rD .t/. The family D will be our attraction universe. Lemma 16.4. Let assumptions .f 1/–.f 2/ and .j1/–.j4/ hold. The non-autonomous set B D fB.t/gt2R defined as B.t/ D fv 2 H W jvj R.t/g is D -pullback absorbing and B 2 Rı . Proof. Obviously R 2 R and hence B 2 D . Take t 2 R and D 2 D . There exists rD 2 R with jwj rD .s/ for all w 2 D.s/ and s 2 R. We can choose t0 D t0 .D; t/ small enough such that rD . /2 e et for all t0 . We will consider a solution u of Problem 16.1 with the initial time t0 and the initial condition u0 2 D. /. From (16.18) by the Poincaré inequality we obtain d ju.s/j2 C . dk k2 /1 ju.s/j2 C1 .1 C jf .s/j2 / dt

for a.e.

s 2 .; t/:


373

From the Gronwall inequality we get ju.t/j2 ju. /j2 e.t / C

C1 C C1 et

Z

t

es jf .s/j2 ds:

(16.23)

Hence ju.t/j2 ju. /j2 e.t / C R2 .t/ 1: Since u. / 2 D. /, then ju.t/j2 rD . /2 e et 1 C R2 .t/: Due to the bound rD . /2 e et , we have ju.t/j2 R2 .t/; whence the assertion follows.

t u

We pass to the next lemma which states that the m-process U is D -pullback asymptotically compact. Lemma 16.5. Assume .f 1/–.f 2/ and .j1/–.j4/. The m-process U is D -asymptotically compact. Proof. The proof uses the technique of Proposition 7.4 in [11] (alternatively it is also possible to obtain the same assertion by the method of [201]) and is based on the energy equation. Let fD.t/gt2R D D 2 D and let uk0 2 D.t0k / with t0k ! 1. Let moreover zk 2 U.t; t0k ; uk0 /. We show that the sequence fzk g is relatively compact in H. There exists a sequence of functions uk 2 L2 .t0k ; t C 1I V/ \ C.Œt0k ; t C 1I H/ with u0n 2 L2 .t0k ; t C 1I V 0 /, solutions of Problem 16.1, such that uk .t0k / D uk0 and uk .t/ D zk . From Lemma 16.4 it follows that there exists N0 such that for all natural k N0 we have juk .t 1/j R.t 1/. From now on we will consider the sequence 2 fuk g1 kDN0 . The selections of [email protected];uk .t// given in (16.12) will be denoted by k . Note that the restrictions of uk ; k to the interval Œt 1; t C 1 solve Problem 16.1 on this interval, with the initial conditions uk .t 1/ taken at t 1. Using (16.18) it follows that the sequence uk is uniformly bounded in L2 .t 1; t C 1I V/ \ C.Œt 1; t C 1I H/, and, by (16.19) it follows that u0k is uniformly bounded in L2 .t 1; t C 1I V 0 /. The growth condition .j3/ implies that k are bounded in L2 .t 1; t C 1I L2 .C //. These bounds are sufficient to extract a subsequence, not renumbered, such that for certain u 2 L2 .t 1; t C 1I V/ with u0 2 .t 1; t C 1I V 0 / and 2 L2 .t 1; t C 1I L2 .C // the following convergences hold:

374


uk ! u

weakly in L2 .t 1; t C 1I V/;

u0k ! u0

weakly in L2 .t 1; t C 1I V 0 /;

uk ! u

strongly in L2 .t 1; t C 1I H/;

(16.24)

uk .s/ ! u.s/

weakly in H

uk ! u

strongly in L2 .t 1; t C 1I L2 .C //;

(16.26)

weakly in L2 .t 1; t C 1I L2 .C //:

(16.27)

k !

s 2 Œt 1; t C 1;

for all

(16.25)

In view of (16.25) it is sufficient to show that juk .t/j ! ju.t/j, whence, as H is a Hilbert space, it follows that un .t/ ! u.t/ strongly in H and the assertion will be proved. Note that by (16.24), for another subsequence, also not renumbered, the strong convergence uk .s/ ! u.s/ holds in H for a.e. s 2 .t 1; t C 1/. Taking the test function uk .t/ in (16.11) written for uk , where the corresponding is denoted by k , and integrating from t 1 to s 2 Œt 1; t C 1, we get 1 juk .s/j2 C 2

Z

s

Z

s

a.uk .r/; uk .r// dr C

t1

1 D juk .t 1/j2 C 2

Z

t1 s

.k .r/; uk .r//L2 .C / dr

.f .r/; uk .r// dr:

(16.28)

t1

We define the functions Vk W Œt 1; t C 1 ! R as Vk .s/ D

1 juk .t 1/j2 2

Z

s

a.uk .r/; uk .r// dr:

t1

Note that the coercivity of a implies that the functions Vk are nonincreasing. By the energy equation (16.28) we have 1 Vk .s/ D juk .s/j2 C 2

Z

Z

s

t1

.k .r/; uk .r//L2 .C / dr

s

.f .r/; uk .r// dr:

t1

From (16.24), (16.26), and (16.27) it follows that lim Vk .s/ D

k!1

1 ju.s/j2 C 2

Z

s

t1

Z ..r/; u .r//L2 .C / dr

s

.f .r/; u.r// dr

t1

for a.e. s 2 .t 1; t C 1/. We denote the right-hand side of the above equation by V.s/. We can choose sequences pm % t and qm & t such that Vk .pm / ! V.pm / and Vk .qm / ! V.qm / as k ! 1 for all m 2 N. We have Vk .pm / Vk .t/ Vk .qm /;


375

and hence V.pm / D lim Vk .pm / lim sup Vk .t/ lim inf Vk .t/ lim Vk .qm / D V.qm /: k!1

k!1

k!1

k!1

We pass with m to infinity and use the fact that V, by definition, is continuous. We get V.t/ lim sup Vk .t/ lim inf Vk .t/ V.t/; k!1

k!1

whence Vk .t/ ! V.t/ as k ! 1. But we have Z

t

t1

Z .k .r/; uk .r//L2 .C / dr Z !

t t1

t

.f .r/; uk .r// dr

t1

..r/; u .r//L2 .C / dr

Z

t

.f .r/; u.r// dr;

t1

which gives us the convergence jun .t/j ! ju.t/j, and the proof is complete.

t u

Lemma 16.6. If for every integer k, uk solves Problem 16.1 with the initial condition u0k taken at t0 , and u0k ! u0 strongly in H then for any t > t0 , for a subsequence, uk .t/ ! u.t/ weakly in H, where u is a solution of Problem 16.1 with the initial condition u0 taken at t0 . In consequence, the m-process U on H satisfies condition (NW). Proof. The proof is standard since the a priori estimates of Lemma 16.3 provide enough convergence to pass to the limit. Indeed, the sequence uk is bounded in L2 .t0 ; tI V/ with u0k bounded in L2 .t0 ; tI V 0 /. Hence, for a subsequence, uk ! u weakly in L2 .t0 ; tI V/ with u0k ! u0 weakly in L2 .t0 ; tI V 0 /. In consequence, for all s 2 Œ0; t, uk .s/ ! u.s/ weakly in H, which means that u.t0 / D u0 and uk .t/ ! u.t/ weakly in H. It also follows that uk ! u strongly in L2 .t0 ; tI L2 .C // and in L2 .C .t0 ; t//. We must show that u solves Problem 16.1 on .t0 ; t/. We only discuss passing to the limit in the multivalued term since for other terms it is standard. Let 2 k 2 L2 .t0 ; tI L2 .C // be such that k .s/ 2 [email protected];u a.e. s 2 .t0 ; t/ and (16.11) holds. k .s// From the growth condition .j3/ it follows that k is bounded in L2 .t0 ; tI L2 .C // and thus, for a subsequence, we have k ! weakly in L2 .t0 ; tI L2 .C // and weakly in L2 .C .t0 ; t//. Using assertion .H2/ in Theorem 3.24 it follows that for a.e. .x; s/ 2 C .t0 ; t/ .x; s/ 2 @j.x; s; u .x; s//, which, in turn, implies that 2 .s/ 2 [email protected];u for a.e. s 2 .t0 ; t/, and the proof is complete. t u .s// From Lemmas 16.4–16.6 it follows that all assumptions of Corollary 16.3 hold and hence we have shown the following Theorem. Theorem 16.4. The m-process U on H associated with Problem 16.1 has a D pullback attractor. This attractor is moreover invariant and it belongs to D .

376


In the above theorem the attractor invariance and the fact that it belongs to the universe D follow from Corollary 16.1 as the m-process is strict and the attraction universe D satisfies .HD1/.

16.3 Comments and Bibliographical Notes In this chapter we were interested in the evolution problems for which the solutions for a given initial conditions are not unique. To deal with such problems, the theory of m-semiflows and their global attractors was introduced in [171, 172] and later extended to non-autonomous case [41]. Results on the existence of pullback D-attractors for the problems without uniqueness of solutions were obtained in [4, 35, 169]. The abstract result on the existence of the pullback D-attractor shown in this chapter was based on these works, however, we showed some extension to the results presented there: our alternative proof of the pullback D-attractor existence was based on the notion of !-D-limit-compactness and seems particularly transparent. We weakened the upper semicontinuity assumption from [4, 35, 169] replacing it by the graph closedness, and we additionally studied not only necessary but also the necessary and sufficient conditions for the attractor existence. We introduced the so-called condition (NW), earlier studied for autonomous case in [240] and recently in [125], and for non-autonomous case in [231] in context of uniform attractors. This condition is convenient to verify for particular problems of mathematical physics governed by PDEs. For the notions of uniform attractors, pullback attractors, and skew-product flows we refer to [10, 53, 55, 130, 131, 239]. For similar problems in contact mechanics, cf. [126, 162] and, in particular, [12, 213]. The results of this chapter come from [127].

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Index

Symbols !-limit set, 152 pullback, 262

-cocycle, 260 continuous, 261 pullback D-asymptotically compact, 261 A attraction universe, 261, 266, 277, 360 inclusion-closed, 266, 364 attractor (global) pullback D-, 261, 278 for m-process, 362 minimal, 262 exponential, 209 global, 151, 208 weak, 334, 339 trajectory, 330 B Banach contraction principle, 42 Banach generalized limit, 170 Bochner integral, 62 boundary condition Fourier, 112 Tresca, 112 Boussinesq equations, 24 C Cauchy equation of motion, 17 principle, 16 coercive bilinear form, 40

complex amplitude, 56 condition (NW), 366 contraction, 42 coordinate material, 12 spatial, 12 Couette flow, 36 D derivative distributional, 48 time, 63 generalized, 46 material, 13 determining modes, 252 determining nodes, 256, 257 differential inclusion, 234 dilation group, 33 dimension fractal, 183, 209 Hausdorff, 193 Dirac delta, 47 directional derivative generalized Clarke, 71 dissipation, function, 23 distribution, 47 regular, 47 Du Bois-Reymond, lemma, 45

E Ehrling lemma, 65 energy dissipation rate, 196

© Springer International Publishing Switzerland 2016 G. Łukaszewicz, P. Kalita, Navier–Stokes Equations, Advances in Mechanics and Mathematics 34, DOI 10.1007/978-3-319-27760-8

387

388 energy transfer direct, 165 inverse, 166 energy, specific internal, 19 enstrophy, 165 entropy, balance of, 27 equation energy, 178, 288 enstrophy, 156 Liouville, 176, 286 stationary, 176 Euler formula, 14 evolutionary system, 338 F field equations, 22 flow large scale component, 164 small scale component, 164 fluid incompressible, 14 Newtonian, 21 non-Newtonian, 21 perfect, 18 polar, 21 Stokesian, 21 force body, 16 contact, 16 Fourier coefficient, 56 Fourier law, 20 friction coefficient, 112 multifunction, 96, 341 function Bochner integrable, 62 locally integrable, 45 weakly continuous, 66 function space Ck .˝/, 45 Ck .˝/, 45 C1 .˝/, 45 C01 .˝/, 45 Cp1 .Q/, 54 Hps .Q/, 54 H m .˝/, 47 1 Lloc .˝/, 45 L1 .˝/, 45 Lp .˝/, 44 p Lloc .˝/, 45 W m;q .˝/, 47 W m;p .˝/, 46 m;p W0 .˝/, 46

Index P ps .Q/, 56 H LP 2 .Q/, 59 D 0 .˝/, 47 D.˝/, 47 G Galerkin method, 40 Galilean invariance, 31 Grashof number, 159, 252 gravity acceleration, 24 H Hausdorff semi-distance, 151, 261, 359 weak, 338 heat flux, 19 I inequality Agmon, 157 first energy, 145 Gronwall, 67 generalized, 70 translated, 69 uniform, 68 Hardy, 53 Korn, 117 Ladyzhenskaya, 51, 218, 343 Lieb–Thirring, 200 Poincaré, 50, 56, 120 second energy, 155 K kinematic viscosity coefficient, 23 Kuratowski measure of noncompactness, 360 Kuratowski–Painlevé upper limit, 74 L l-trajectory, 209 law of thermodynamics first, 19 second, 27 Lax–Milgram, lemma, 40 local variable, 25 M measure invariant, 169 regular, 175

Index N Navier–Stokes equations, 24 nodal point, 256 O operator completely continuous, 42 P point nonwandering, 172 wandering, 173 Poiseuille flow, 35 pressure, 18 principle Banach contraction, 42 dimensional homogenity, 33 of conservation of linear momentum, 16 of conservation of mass, 15 process evolutionary, 266, 297 -continuous, 291 pullback D-asymptotically compact, 266 multivalued (m-process), 360 closed, 362 pullback !-D-limit compact, 360 pullback D-asymptotically compact, 360 strict, 360 R Reynolds equation, 141 weak form, 137 number, 29 rotation symmetry, 31 S scaling, 31 semiflow uniformly quasidifferentiable, 186 semigroup, 151 asymptotically compact, 154 continuous, 151 uniformly compact, 152 set absorbing, 152 for shift semigroup, 330 attracting for shift semigroup, 330 weakly, 338

389 invariant, 151, 339 non-autonomous, 261, 266, 360 pullback D-absorbing, 267, 362 parameterized, 261 pullback D-absorbing, 261 pullback D-attracting, 261 prevalent, 185 similarity dynamical, 29 variable, 33 smoothing estimate, 191 Sobolev space, 46 solution approximate, 41, 144 invariant, 32 Leray–Hopf, 347, 353 self-similar, 31 statistical, 289 stationary, 178, 290 squeezing property, 192 Stokes operator, 84, 86 problem, 84 stress hydrostatic, 18 normal, 16 subdifferential generalized Clarke, 71 support of function, 45 of measure, 172 symmetry group, 31 T tensor deformation, 20 stress, 17 stress, viscous, 18 theorem Aubin–Lions, 66 Brouwer, 42 Hölder–Mañé, 185 Hahn–Banach, 175 Kakutani–Fan–Glicksberg, 44 Lebourg mean value, 72 Leray–Schauder, 43 of stress means, 19 Rellich, 49 Riesz–Fréchet, 41 Riesz–Kakutani, 175 Schauder, 42 Tietze, 176 transport, 13

390 theory false, 37 incomplete, 37 overdetermined, 37 well-set, 36 thermal conductivity, 20 diffusion coefficient, 24 expansion coefficient, 24 tracking property, 339 Tresca condition, 211 generalized, 233, 367

Index V variable hydromechanical, 12 state, 26 variational inequality, 215 velocity, 12 vorticity, 25

W wave number, 56

Navier–Stokes Equations

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