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

Description

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. Cover Title Copyright Contents Preface I Introduction 1 The Euler-Reynolds System II General Considerations of the Scheme 2 Structure of the Book 3 Basic Technical Outline III Basic Construction of the Correction 4 Notation 5 A Main Lemma for Continuous Solutions 6 The Divergence Equation 6.1 A Remark about Momentum Conservation 6.2 The Parametrix 6.3 Higher Order Parametrix Expansion 6.4 An Inverse for Divergence 7 Constructing the Correction 7.1 Transportation of the Phase Functions 7.2 The High-High Interference Problem and Beltrami Flows 7.3 Eliminating the Stress 7.3.1 The Approximate Stress Equation 7.3.2 The Stress Equation and the Initial Phase Directions 7.3.3 The Index Set, the Cutoffs and the Phase Functions 7.3.4 Localizing the Stress Equation 7.3.5 Solving the Quadratic Equation 7.3.6 The Renormalized Stress Equation in Scalar Form 7.3.7 Summary IV Obtaining Solutions from the Construction 8 Constructing Continuous Solutions 8.1 Step 1: Mollifying the Velocity 8.2 Step 2: Mollifying the Stress 8.3 Step 3: Choosing the Lifespan 8.4 Step 4: Bounds for the New Stress 8.5 Step 5: Bounds for the Corrections 8.6 Step 6: Control of the Energy Increment 9 Frequency and Energy Levels 10 The Main Iteration Lemma 10.1 Frequency Energy Levels for the Euler-Reynolds Equations 10.2 Statement of the Main Lemma 11 Main Lemma Implies the Main Theorem 11.1 The Base Case 11.2 The Main Lemma Implies the Main Theorem 11.2.1 Choosing the Parameters 11.2.2 Choosing the Energies 11.2.3 Regularity of the Velocity Field 11.2.4 Asymptotics for the Parameters 11.2.5 Regularity of the Pressure 11.2.6 Compact Support in Time 11.2.7 Nontriviality of the Solution 12 Gluing Solutions 13 On Onsager's Conjecture 13.1 Higher Regularity for the Energy V Construction of Regular Weak Solutions: Preliminaries 14 Preparatory Lemmas 15 The Coarse Scale Velocity 16 The Coarse Scale Flow and Commutator Estimates 17 Transport Estimates 17.1 Stability of the Phase Functions 17.2 Relative Velocity Estimates 17.3 Relative Acceleration Estimates 18 Mollification along the Coarse Scale Flow 18.1 The Problem of Mollifying the Stress in Time 18.2 Mollifying the Stress in Space and Time 18.3 Choosing Mollification Parameters 18.4 Estimates for the Coarse Scale Flow 18.5 Spatial Variations of the Mollified Stress 18.6 Transport Estimates for the Mollified Stress 18.6.1 Derivatives and Averages along the Flow Commute 18.6.2 Material Derivative Bounds for the Mollified Stress 18.6.3 Second Time Derivative of the Mollified Stress along the Coarse Scale Flow 18.6.4 An Acceptability Check 19 Accounting for the Parameters and the Problem with the High-High Term VI Construction of Regular Weak Solutions: Estimating the Correction 20 Bounds for Coefficients from the Stress Equation 21 Bounds for the Vector Amplitudes 22 Bounds for the Corrections 22.1 Bounds for the Velocity Correction 22.2 Bounds for the Pressure Correction 23 Energy Approximation 24 Checking Frequency Energy Levels for the Velocity and Pressure VII Construction of Regular Weak Solutions: Estimating the New Stress 25 Stress Terms Not Involving Solving the Divergence Equation 25.1 The Mollification Term from the Velocity 25.2 The Mollification Term from the Stress 25.3 Estimates for the Stress Term 26 Terms Involving the Divergence Equation 26.1 Expanding the Parametrix 26.2 Applying the Parametrix 27 Transport-Elliptic Estimates 27.1 Existence of Solutions for the Transport-Elliptic Equation 27.2 Spatial Derivative Estimates for the Solution to the Transport- Elliptic Equation 27.3 Material Derivative Estimates for the Transport-Elliptic Equation 27.4 Cutting Off the Solution to the Transport-Elliptic Equation Acknowledgments Appendices A The Positive Direction of Onsager's Conjecture B Simplifications and Recent Developments References Index