Lecture Notes on Regularity Theory for the Navier-Stokes Equations

269 Pages·2014·1.464 MB·English

by  Gregory Seregin

LECTURE NOTES ON
REGULARITY THEORY FOR THE
NAVIER-STOKES EQUATIONS
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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
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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.
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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.
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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
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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
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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