- 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 Main Iteration Lemma
The Main Iteration Lemma
- Chapter:
- 10 The Main Iteration Lemma
- Source:
- Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time
- Author(s):
Philip Isett
- Publisher:
- Princeton University Press
This chapter properly formalizes the Main Lemma, first by discussing the frequency energy levels for the Euler-Reynolds equations. Here the bounds are all consistent with the symmetries of the Euler equations, and the scaling symmetry is reflected by dimensional analysis. The chapter proceeds by making assumptions that are consistent with the Galilean invariance of the Euler equations and the Euler-Reynolds equations. If (v, p, R) solve the Euler-Reynolds equations, then a new solution to Euler-Reynolds with the same frequency energy levels can be obtained. The chapter also states the Main Lemma, taking into account dimensional analysis, energy regularity, and Onsager's conjecture. Finally, it introduces the main theorem (Theorem 10.1), which states that there exists a nonzero solution to the Euler equations with compact support in time.
Keywords: frequency energy levels, Euler-Reynolds equations, scaling symmetry, dimensional analysis, Euler equations, Main Lemma, energy regularity, Onsager's conjecture, nonzero solution
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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
