Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

214 Pages·2017·1.347 MB·English

by  Philip Isett

Annals of Mathematics Studies
Number 196

Ho¨lder Continuous Euler Flows in Three Dimensions
with Compact Support in Time
Philip Isett
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
2017
Copyright (cid:13)c 2017 by Princeton University Press
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All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Names: Isett, Philip, 1986–
Title: H¨older continuous Euler flows in three dimensions with compact
support in time / Philip Isett.
Description: Princeton: PrincetonUniversityPress,[2017]|Series: Annals
of mathematics studies ; number 196 | Author’s thesis
(doctoral)—Princeton University, Princeton, N.J., 2013 | Includes
bibliographical references and index.
Identifiers: LCCN 2016042604| ISBN 9780691174822 (hardcover : alk.
paper) | ISBN 9780691174839 (pbk. : alk. paper)
Subjects: LCSH: Fluid dynamics—Mathematics.
Classification: LCC QA911 .I84 2017 | DDC 532/.05dc23
LC record available at https://lccn.loc.gov/2016042604
British Library Cataloging-in-Publication Data is available
The publisher would like to acknowledge the author of this volume for
providing the camera-ready copy from which this book was printed.
This book has been composed in LaTeX
Printed on acid-free paper ∞
10 9 8 7 6 5 4 3 2 1
Contents
Preface ix
I Introduction 1
1 The Euler-Reynolds System 7
II General Considerations of the Scheme 11
2 Structure of the Book 13
3 Basic Technical Outline 14
III Basic Construction of the Correction 19
4 Notation 19
5 A Main Lemma for Continuous Solutions 20
6 The Divergence Equation 22
6.1 A Remark about Momentum Conservation 22
6.2 The Parametrix 24
6.3 Higher Order Parametrix Expansion 27
6.4 An Inverse for Divergence 28
7 Constructing the Correction 30
7.1 Transportation of the Phase Functions 30
7.2 The High-High Interference Problem and Beltrami Flows 32
7.3 Eliminating the Stress 36
7.3.1 The Approximate Stress Equation 36
7.3.2 The Stress Equation and the Initial Phase Directions 38
7.3.3 The Index Set, the Cutoffs and the Phase Functions 40
7.3.4 Localizing the Stress Equation 45
7.3.5 Solving the Quadratic Equation 46
7.3.6 The Renormalized Stress Equation in Scalar Form 50
7.3.7 Summary 54
IV Obtaining Solutions from the Construction 56
CONTENTS
8 Constructing Continuous Solutions 56
8.1 Step 1: Mollifying the Velocity 56
8.2 Step 2: Mollifying the Stress 57
8.3 Step 3: Choosing the Lifespan 58
8.4 Step 4: Bounds for the New Stress 59
8.5 Step 5: Bounds for the Corrections 60
8.6 Step 6: Control of the Energy Increment 60
9 Frequency and Energy Levels 62
10 The Main Iteration Lemma 67
10.1 Frequency Energy Levels for the Euler-Reynolds Equations 67
10.2 Statement of the Main Lemma 68
11 Main Lemma Implies the Main Theorem 71
11.1 The Base Case 72
11.2 The Main Lemma Implies the Main Theorem 75
11.2.1 Choosing the Parameters 75
11.2.2 Choosing the Energies 77
11.2.3 Regularity of the Velocity Field 78
11.2.4 Asymptotics for the Parameters 81
11.2.5 Regularity of the Pressure 85
11.2.6 Compact Support in Time 86
11.2.7 Nontriviality of the Solution 87
12 Gluing Solutions 89
13 On Onsager’s Conjecture 90
13.1 Higher Regularity for the Energy 92
V Construction of Regular Weak Solutions:
Preliminaries 97
14 Preparatory Lemmas 97
15 The Coarse Scale Velocity 99
16 The Coarse Scale Flow and Commutator Estimates 102
17 Transport Estimates 105
17.1 Stability of the Phase Functions 106
17.2 Relative Velocity Estimates 108
17.3 Relative Acceleration Estimates 113
vi
CONTENTS
18 Mollification along the Coarse Scale Flow 115
18.1 The Problem of Mollifying the Stress in Time 115
18.2 Mollifying the Stress in Space and Time 116
18.3 Choosing Mollification Parameters 116
18.4 Estimates for the Coarse Scale Flow 119
18.5 Spatial Variations of the Mollified Stress 122
18.6 Transport Estimates for the Mollified Stress 123
18.6.1 Derivatives and Averages along the Flow Commute 124
18.6.2 Material Derivative Bounds for the Mollified Stress 126
18.6.3 Second Time Derivative of the Mollified Stress along
the Coarse Scale Flow 129
18.6.4 An Acceptability Check 130
19 Accounting for the Parameters and the Problem with the
High-High Term 131
VI Construction of Regular Weak Solutions:
Estimating the Correction 135
20 Bounds for Coefficients from the Stress Equation 135
21 Bounds for the Vector Amplitudes 138
22 Bounds for the Corrections 143
22.1 Bounds for the Velocity Correction 143
22.2 Bounds for the Pressure Correction 146
23 Energy Approximation 147
24 Checking Frequency Energy Levels for the Velocity and
Pressure 150
VII Construction of Regular Weak Solutions:
Estimating the New Stress 152
25 Stress Terms Not Involving Solving the Divergence
Equation 155
25.1 The Mollification Term from the Velocity 156
25.2 The Mollification Term from the Stress 161
25.3 Estimates for the Stress Term 162
26 Terms Involving the Divergence Equation 163
26.1 Expanding the Parametrix 164
26.2 Applying the Parametrix 168
vii
CONTENTS
27 Transport-Elliptic Estimates 173
27.1 Existence of Solutions for the Transport-Elliptic Equation 176
27.2 SpatialDerivativeEstimatesfortheSolutiontotheTransport-
Elliptic Equation 179
27.3 Material Derivative Estimates for the Transport-Elliptic
Equation 181
27.4 Cutting Off the Solution to the Transport-Elliptic Equation 182
Acknowledgments 183
Appendices 185
A The Positive Direction of Onsager’s Conjecture 191
B Simplifications and Recent Developments 194
References 197
Index 201
viii
Preface
ThepresentmonographconcernstheconstructionofH¨oldercontinuous
weak solutions to the incompressible Euler equations that fail to conserve
energy. ThismanuscriptoriginallyappearedasanarXivpreprintinNovem-
ber2012, andevolvedintotheauthor’sPh.D.thesisinthePrincetonmath
department [Ise13b]. The greatest motivation for constructing such solu-
tions has come from the pursuit of Onsager’s conjecture, which states that
for any α < 1/3 there exist weak solutions to incompressible Euler with
H¨older exponent α for which the total kinetic energy fails to be conserved.
In the following we show the existence of (1/5−(cid:15))-H¨older Euler flows that
failtoconserveenergy,whichisatthistimethebestresulttowardsproving
Onsager’s conjecture in its original formulation.
Afewyearshavepassedsincethefirstappearanceofthiswork,andthe
study of weak solutions to incompressible Euler and related equations has
seen many developments in this time. To provide an account closer to the
stateoftheartofthisstillveryactivefieldofresearch,weprovideherealist
of some references to recent work and comment briefly on the relationship
to the present work. We will present some more detailed discussion in the
Appendix Section (B).
Shortlyfollowingtheappearanceofthepresentwork,asecondconstruc-
tionof(1/5−(cid:15))-H¨olderEulerflowswasgivenbyBuckmaster,DeLellisand
Sz´ekelyhidi [BDLS13]. Their main result shows also that the kinetic en-
ergy profile may be prescribed to be any strictly positive smooth function.
The proof given in [BDLS13] is much shorter than the one given here, and
adheres more closely to the previous work of [DLS13, DLS14]. We refer to
[BDLIS15] for a self-contained presentation of the proof in [BDLS13] that
includes a more detailed technical comparison to the present work.
In [Buc15, BDLS15], Buckmaster, De Lellis and Sz´ekelyhidi have con-
structed Euler flows with Onsager critical spatial regularity in the class
C0 ∩L1C1/3−(cid:15). We note that this result has required some technical de-
t,x t x
parturesfromtheconstructiongivenhere. Wealsonotethattheargument
of [BDLS15] makes crucial use of the technique of “mollification along the
flow” introduced below, whereas in [BDLS13] this technique had been re-
placed by a simpler alternative.
In [Ise13a], the author presented a theory of time regularity for Euler
flowsintheclassL∞Cα,andproposedastrongerformofOnsager’sconjec-
t x
ture called the “Generic Failure of Energy Regularity,” which is elaborated
in[IOa]. Anextensionofthepresentconstructiontoproducesolutionswith
compact support in space and time is given in [IOa, IOb], which contain
also some first steps towards proving the conjectured “Generic Failure of
Energy Regularity” for solutions with H¨older exponent α<1/5. A version
of the present construction has also been extended to a general class of
active scalar equations in [IV15], where an analogue of Onsager’s conjec-