- 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
Main Lemma Implies the Main Theorem
Main Lemma Implies the Main Theorem
- Chapter:
- 11 Main Lemma Implies the Main Theorem
- Source:
- Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time
- Author(s):
Philip Isett
- Publisher:
- Princeton University Press
This chapter shows that the Main Lemma implies the main theorem. It proves Theorem (10.1) by inductively applying the Main Lemma in order to construct a sequence of solutions of the Euler-Reynolds system. At each stage of the induction, an energy function is chosen along with a parameter whose choice determines the growth of the frequency parameter and the decay of the energy level. A base case lemma is then established, after which the proof of the Main Theorem (10.1) is presented so that the Main Lemma implies the Main Theorem. The Main Lemma is employed to approximately prescribe the energy increment of the correction. The solution obtained at the end of the process is nontrivial.
Keywords: correction, Main Lemma, Euler-Reynolds system, energy function, Main Theorem, energy increment
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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
