- 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
Mollification along the Coarse Scale Flow
Mollification along the Coarse Scale Flow
- Chapter:
- 18 Mollification along the Coarse Scale Flow
- Source:
- Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time
- Author(s):
Philip Isett
- Publisher:
- Princeton University Press
This chapter shows how to construct the appropriate mollification of the Reynolds stress along the coarse scale flow. Unlike the velocity field, which was only mollified in the spatial variables and which earned its time-regularity through the Euler-Reynolds equation, the Reynolds stress must be mollified in both space and time. Mollification along the flow is consistent with the Galilean invariance of the equations. After considering the problem of mollifying the stress in time, the chapter explains how the stress can be mollified in both space and time. It then chooses the mollification parameters, requiring that the error term generated by this mollification constitutes a small fraction of the allowable stress. It also derives estimates for the coarse scale flow as well as transport estimates for the mollified stress.
Keywords: mollification, Reynolds stres, coarse scale flow, Galilean invariance, error term, transport estimate
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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
