- 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
Transport-Elliptic Estimates
Transport-Elliptic Estimates
- Chapter:
- 27 Transport-Elliptic Estimates
- Source:
- Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time
- Author(s):
Philip Isett
- Publisher:
- Princeton University Press
This chapter solves the underdetermined, elliptic equation ∂ⱼQsuperscript jl = Usuperscript l and Qsuperscript jl = Qsuperscript lj (Equation 1069) in order to eliminate the error term in the parametrix. For the proof of the Main Lemma, estimates for Q and the material derivative as well as its spatial derivatives are derived. The chapter finds a solution to Equation (1069) with good transport properties by solving it via a Transport equation obtained by commuting the divergence operator with the material derivative. It concludes by showing the solutions, spatial derivative estimates, and material derivative estimates for the Transport-Elliptic equation, as well as cutting off the solution to the Transport-Elliptic equation.
Keywords: error term, parametrix, Main Lemma, material derivative, spatial derivative, Transport equation, divergence operator, Transport-Elliptic 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
