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

by Philip Isett

Overview

Motivated by the theory of turbulence in fluids, the physicist and chemist Lars Onsager conjectured in 1949 that weak solutions to the incompressible Euler equations might fail to conserve energy if their spatial regularity was below 1/3-Hölder. In this book, Philip Isett uses the method of convex integration to achieve the best-known results regarding nonuniqueness of solutions and Onsager's conjecture. Focusing on the intuition behind the method, the ideas introduced now play a pivotal role in the ongoing study of weak solutions to fluid dynamics equations.

The construction itself—an intricate algorithm with hidden symmetries—mixes together transport equations, algebra, the method of nonstationary phase, underdetermined partial differential equations (PDEs), and specially designed high-frequency waves built using nonlinear phase functions. The powerful "Main Lemma"—used here to construct nonzero solutions with compact support in time and to prove nonuniqueness of solutions to the initial value problem—has been extended to a broad range of applications that are surveyed in the appendix. Appropriate for students and researchers studying nonlinear PDEs, this book aims to be as robust as possible and pinpoints the main difficulties that presently stand in the way of a full solution to Onsager's conjecture.

Product Details

ISBN-13:	9780691174839
Publisher:	Princeton University Press
Publication date:	02/21/2017
Series:	Annals of Mathematics Studies , #196
Pages:	216
Product dimensions:	6.00(w) x 9.20(h) x 0.40(d)

About the Author

Philip Isett is assistant professor of mathematics at the University of Texas, Austin.

Table of 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

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

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 Corrections 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

27 Transport-Elliptic Estimates 173

27.1 Existence of Solutions for the Transport-Elliptic Equation 176

27.2 Spatial Derivative Estimates for the Solution to the Transport-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