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

(ams-196)
 
 
Princeton University Press
  • 1. Auflage
  • |
  • erschienen am 21. Februar 2017
  • |
  • 216 Seiten
 
E-Book | PDF mit Adobe-DRM | Systemvoraussetzungen
978-1-4008-8542-8 (ISBN)
 

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.

  • Englisch
  • Princeton
  • |
  • USA
  • Für Beruf und Forschung
  • |
  • US School Grade: College Graduate Student
  • Digitale Ausgabe
  • 1,38 MB
978-1-4008-8542-8 (9781400885428)
weitere Ausgaben werden ermittelt
Philip Isett
  • 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

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