 Title Pages
 Preface
 Introduction

1 The EulerReynolds 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 HighHigh 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 TransportElliptic Estimates  Appendices
 References
 Index
TransportElliptic Estimates
TransportElliptic Estimates
 Chapter:
 27 TransportElliptic 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 TransportElliptic equation, as well as cutting off the solution to the TransportElliptic equation.
Keywords: error term, parametrix, Main Lemma, material derivative, spatial derivative, Transport equation, divergence operator, TransportElliptic equation
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 Title Pages
 Preface
 Introduction

1 The EulerReynolds 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 HighHigh 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 TransportElliptic Estimates  Appendices
 References
 Index