- Title Pages
- Preface
- Introduction
- 1 The Euler-Reynolds System
- Part II General Considerations of the Scheme
- 2 Structure of the Book
- 3 Basic Technical Outline
- 4 Notation
- 5 A Main Lemma for Continuous Solutions
- 6 The Divergence Equation
- 7 Constructing the Correction
- 8 Constructing Continuous Solutions
- 9 Frequency and Energy Levels
- 10 The Main Iteration Lemma
- 11 Main Lemma Implies the Main Theorem
- 12 Gluing Solutions
- 13 On Onsager's Conjecture
- 14 Preparatory Lemmas
- 15 The Coarse Scale Velocity
- 16 The Coarse Scale Flow and Commutator Estimates
- 17 Transport Estimates
- 18 Mollification along the Coarse Scale Flow
- 19 Accounting for the Parameters and the Problem with the High-High Term
- Part 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
- 23 Energy Approximation
- 24 Checking Frequency Energy Levels for the Velocity and Pressure
- Part VII Construction of Regular Weak Solutions: Estimating the New Stress
- 25 Stress Terms Not Involving Solving the Divergence Equation
- 26 Terms Involving the Divergence Equation
- 27 Transport-Elliptic Estimates
- Appendices
- References
- Index

# The Euler-Reynolds System

# The Euler-Reynolds System

- Chapter:
- 1 The Euler-Reynolds System
- Source:
- Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time
- Author(s):
### Philip Isett

- Publisher:
- Princeton University Press

This chapter provides a background on the Euler-Reynolds system, starting with some of the underlying philosophy behind the argument. It describes low frequency parts and ensemble averages of Euler flows and shows that the average of any family of solutions to Euler will be a solution of the Euler-Reynolds equations. It explains how the most relevant type of averaging to convex integration arises during the operation of taking weak limits, which can be regarded as an averaging process. The chapter proceeds by focusing on weak limits of Euler flows and the hierarchy of frequencies, concluding with a discussion of the method of convex integration and the *h*-principle for weak limits. The method inherently proves that weak solutions to Euler may fail to be solutions.

*Keywords:*
weak solution, Euler flow, Euler-Reynolds equations, convex integration, weak limit, frequencies, h-principle

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- Title Pages
- Preface
- Introduction
- 1 The Euler-Reynolds System
- Part II General Considerations of the Scheme
- 2 Structure of the Book
- 3 Basic Technical Outline
- 4 Notation
- 5 A Main Lemma for Continuous Solutions
- 6 The Divergence Equation
- 7 Constructing the Correction
- 8 Constructing Continuous Solutions
- 9 Frequency and Energy Levels
- 10 The Main Iteration Lemma
- 11 Main Lemma Implies the Main Theorem
- 12 Gluing Solutions
- 13 On Onsager's Conjecture
- 14 Preparatory Lemmas
- 15 The Coarse Scale Velocity
- 16 The Coarse Scale Flow and Commutator Estimates
- 17 Transport Estimates
- 18 Mollification along the Coarse Scale Flow
- 19 Accounting for the Parameters and the Problem with the High-High Term
- Part 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
- 23 Energy Approximation
- 24 Checking Frequency Energy Levels for the Velocity and Pressure
- Part VII Construction of Regular Weak Solutions: Estimating the New Stress
- 25 Stress Terms Not Involving Solving the Divergence Equation
- 26 Terms Involving the Divergence Equation
- 27 Transport-Elliptic Estimates
- Appendices
- References
- Index