LECTURE NOTES ON REGULARITY THEORY FOR THE NAVIER-STOKES EQUATIONS 9314_9789814623407_tp.indd 1 1/9/14 2:17 pm May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk LECTURE NOTES ON REGULARITY THEORY FOR THE NAVIER-STOKES EQUATIONS Gregory Seregin Oxford University, UK St. Petersburg Department of Steklov Mathematical Institute, RAS, Russia World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 9314_9789814623407_tp.indd 2 1/9/14 2:17 pm Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Seregin, Gregory, 1950– author. [Lecture notes. Selections] Lecture notes on regularity theory for the Navier-Stokes equations / Gregory Seregin (Oxford University, UK). pages cm Includes bibliographical references and index. ISBN 978-9814623407 (hardcover : alk. paper) 1. Navier-Stokes equations. 2. Fluid dynamics. I. Title. QA377.S463 2014 515'.353--dc23 2014024553 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2015 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore RokTing - Lec Notes on Regularity Theory.indd 1 10/7/2014 2:43:18 PM August27,2014 14:25 LectureNotesonLocalRegularity LectureNotes pagev Preface TheLectureNotesarebasedontheTCC(GraduateTaughtCourseCenter) coursegivenbymeinTrinityTermsof2009-2011atMathematicalInstitute of Oxford University. Chapters 1-3 contain material discussed in Trinity Term of2009(16 hoursin total), Chapters 4-5containlectures of 2010(16 hours), and, finally, lectures of 2011 are covered by Chapter 6 (16 hours). Chapters 1-5 can be regarded as an Introduction to the Mathemati- cal Theory of the Navier-Stokes equations, relying mainly on the classical PDE’s approach. First, the notion of weak solutions is introduced, then their existence is proven(where it is possible), and, afterwards,differentia- bility properties are analyzed. In other words, we treat the Navier-Stokes equationsasaparticularcase,maybeverydifficult,ofthetheoryofnonlin- ear PDE’s. From this point of view, the Lectures Notes do not pretend to be a complete mathematical theory of the Navier-Stokes equations. There are different approaches, for example, more related to harmonic analysis, etc. A corresponding list of references (incomplete, of course) is given at the end of the Lecture Notes. Finally,Chapters6and7containmoreadvancedmaterial,whichreflects my scientific interests. I also would like to thank Tim Shilkin for careful reading of Lecture Notes and for his valuable suggestions. v May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk September1,2014 13:40 LectureNotesonLocalRegularity LectureNotes pagevii Contents Preface v 1. Preliminaries 1 1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Newtonian Potential . . . . . . . . . . . . . . . . . . . . . 5 1.3 Equation divu=b . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Neˇcas Imbedding Theorem. . . . . . . . . . . . . . . . . . 16 1.5 Spaces of Solenoidal Vector Fields . . . . . . . . . . . . . 21 1.6 Linear Functionals Vanishing on Divergence Free Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.7 Helmholtz-Weyl Decomposition . . . . . . . . . . . . . . . 26 1.8 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2. Linear Stationary Problem 31 2.1 Existence and Uniqueness of Weak Solutions . . . . . . . 31 2.2 Coercive Estimates . . . . . . . . . . . . . . . . . . . . . 33 2.3 Local Regularity . . . . . . . . . . . . . . . . . . . . . . . 36 2.4 Further Local Regularity Results, n=2,3 . . . . . . . . . 37 2.5 Stokes Operator in Bounded Domains . . . . . . . . . . . 41 2.6 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3. Non-Linear Stationary Problem 47 3.1 Existence of Weak Solutions . . . . . . . . . . . . . . . . . 47 3.2 Regularity of Weak Solutions . . . . . . . . . . . . . . . . 52 3.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 vii September1,2014 13:40 LectureNotesonLocalRegularity LectureNotes pageviii viii Lecture Noteson Regularity Theory for the Navier-StokesEquations 4. Linear Non-Stationary Problem 61 4.1 Derivative in Time . . . . . . . . . . . . . . . . . . . . . . 61 4.2 Explicit Solution . . . . . . . . . . . . . . . . . . . . . . . 64 4.3 Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . 75 4.4 Pressure Field. Regularity . . . . . . . . . . . . . . . . . . 76 4.5 Uniqueness Results . . . . . . . . . . . . . . . . . . . . . . 80 4.6 Local Interior Regularity . . . . . . . . . . . . . . . . . . . 84 4.7 Local Boundary Regularity . . . . . . . . . . . . . . . . . 88 4.8 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5. Non-linear Non-Stationary Problem 91 5.1 Compactness Results for Non-Stationary Problems . . . . 91 5.2 Auxiliary Problem . . . . . . . . . . . . . . . . . . . . . . 94 5.3 Weak Leray-HopfSolutions . . . . . . . . . . . . . . . . . 101 5.4 Multiplicative Inequalities and Related Questions . . . . . 106 5.5 Uniqueness of Weak Leray-Hopf Solutions. 2D Case . . . 109 5.6 Further Properties of Weak Leray-HopfSolutions . . . . . 114 5.7 Strong Solutions . . . . . . . . . . . . . . . . . . . . . . . 119 5.8 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6. Local Regularity Theory for Non-Stationary Navier- Stokes Equations 133 6.1 ε-Regularity Theory . . . . . . . . . . . . . . . . . . . . . 133 6.2 Bounded Ancient Solutions . . . . . . . . . . . . . . . . . 149 6.3 Mild Bounded Ancient Solutions . . . . . . . . . . . . . . 158 6.4 Liouville Type Theorems . . . . . . . . . . . . . . . . . . 166 6.4.1 LPS Quantities . . . . . . . . . . . . . . . . . . . 166 6.4.2 2D case . . . . . . . . . . . . . . . . . . . . . . . . 167 6.4.3 Axially Symmetric Case with No Swirl . . . . . . 170 6.4.4 Axially Symmetric Case. . . . . . . . . . . . . . . 173 6.5 Axially Symmetric Suitable Weak Solutions . . . . . . . . 178 6.6 BackwardUniqueness for Navier-Stokes Equations . . . . 184 6.7 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7. Behavior of L -Norm 189 3 7.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.2 Estimates of Scaled Solutions . . . . . . . . . . . . . . . . 191 7.3 Limiting Procedure . . . . . . . . . . . . . . . . . . . . . . 197 September1,2014 13:40 LectureNotesonLocalRegularity LectureNotes pageix Contents ix 7.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Appendix A BackwardUniqueness and Unique Continuation 205 A.1 Carleman-Type Inequalities . . . . . . . . . . . . . . . . . 205 A.2 Unique Continuation Across Spatial Boundaries . . . . . . 210 A.3 BackwardUniqueness for Heat Operator in Half Space . . 214 A.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Appendix B Lemarie-Riesset Local Energy Solutions 221 B.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . 221 B.2 Proof of Theorem 1.6. . . . . . . . . . . . . . . . . . . . . 225 B.3 Regularized Problem . . . . . . . . . . . . . . . . . . . . . 233 B.4 Passing to Limit and Proof of Proposition 1.8 . . . . . . . 237 B.5 Proof of Theorem 1.7. . . . . . . . . . . . . . . . . . . . . 243 B.6 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 B.7 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Bibliography 251 Index 257