- 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 Coarse Scale Velocity

# The Coarse Scale Velocity

- Chapter:
- 15 The Coarse Scale Velocity
- Source:
- Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time
- Author(s):
### Philip Isett

- Publisher:
- Princeton University Press

This chapter deals with the coarse scale velocity. It begins the proof of Lemma (10.1) by choosing a double mollification for the velocity field. Here ∈ᵥ is taken to be as large as possible so that higher derivatives of *v*element are less costly, and each *v*subscript Element has frequency smaller than λ so element*v*⁻¹ must be smaller than λ in order of magnitude. Each derivative of *v*subscript Element up to order *L* costs a factor of Ξ. The chapter proceeds by describing the basic building blocks of the construction, the choice of element*v* and the parametrix expansion for the divergence equation.

*Keywords:*
coarse scale velocity, mollification, velocity field, parametrix, divergence equation

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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