Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

Cover
Title
Copyright
Contents
Preface
I Introduction
	1 The Euler-Reynolds System
II General Considerations of the Scheme
	2 Structure of the Book
	3 Basic Technical Outline
III Basic Construction of the Correction
	4 Notation
	5 A Main Lemma for Continuous Solutions
	6 The Divergence Equation
		6.1 A Remark about Momentum Conservation
		6.2 The Parametrix
		6.3 Higher Order Parametrix Expansion
		6.4 An Inverse for Divergence
	7 Constructing the Correction
		7.1 Transportation of the Phase Functions
		7.2 The High-High Interference Problem and Beltrami Flows
		7.3 Eliminating the Stress
			7.3.1 The Approximate Stress Equation
			7.3.2 The Stress Equation and the Initial Phase Directions
			7.3.3 The Index Set, the Cutoffs and the Phase Functions
			7.3.4 Localizing the Stress Equation
			7.3.5 Solving the Quadratic Equation
			7.3.6 The Renormalized Stress Equation in Scalar Form
			7.3.7 Summary
IV Obtaining Solutions from the Construction
	8 Constructing Continuous Solutions
		8.1 Step 1: Mollifying the Velocity
		8.2 Step 2: Mollifying the Stress
		8.3 Step 3: Choosing the Lifespan
		8.4 Step 4: Bounds for the New Stress
		8.5 Step 5: Bounds for the Corrections
		8.6 Step 6: Control of the Energy Increment
	9 Frequency and Energy Levels
	10 The Main Iteration Lemma
		10.1 Frequency Energy Levels for the Euler-Reynolds Equations
		10.2 Statement of the Main Lemma
	11 Main Lemma Implies the Main Theorem
		11.1 The Base Case
		11.2 The Main Lemma Implies the Main Theorem
			11.2.1 Choosing the Parameters
			11.2.2 Choosing the Energies
			11.2.3 Regularity of the Velocity Field
			11.2.4 Asymptotics for the Parameters
			11.2.5 Regularity of the Pressure
			11.2.6 Compact Support in Time
			11.2.7 Nontriviality of the Solution
	12 Gluing Solutions
	13 On Onsager\'s Conjecture
		13.1 Higher Regularity for the Energy
V Construction of Regular Weak Solutions: Preliminaries
	14 Preparatory Lemmas
	15 The Coarse Scale Velocity
	16 The Coarse Scale Flow and Commutator Estimates
	17 Transport Estimates
		17.1 Stability of the Phase Functions
		17.2 Relative Velocity Estimates
		17.3 Relative Acceleration Estimates
	18 Mollification along the Coarse Scale Flow
		18.1 The Problem of Mollifying the Stress in Time
		18.2 Mollifying the Stress in Space and Time
		18.3 Choosing Mollification Parameters
		18.4 Estimates for the Coarse Scale Flow
		18.5 Spatial Variations of the Mollified Stress
		18.6 Transport Estimates for the Mollified Stress
			18.6.1 Derivatives and Averages along the Flow Commute
			18.6.2 Material Derivative Bounds for the Mollified Stress
			18.6.3 Second Time Derivative of the Mollified Stress along the Coarse Scale Flow
			18.6.4 An Acceptability Check
	19 Accounting for the Parameters and the Problem with the High-High Term
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
		22.1 Bounds for the Velocity Correction
		22.2 Bounds for the Pressure Correction
	23 Energy Approximation
	24 Checking Frequency Energy Levels for the Velocity and Pressure
VII Construction of Regular Weak Solutions: Estimating the New Stress
	25 Stress Terms Not Involving Solving the Divergence Equation
		25.1 The Mollification Term from the Velocity
		25.2 The Mollification Term from the Stress
		25.3 Estimates for the Stress Term
	26 Terms Involving the Divergence Equation
		26.1 Expanding the Parametrix
		26.2 Applying the Parametrix
	27 Transport-Elliptic Estimates
		27.1 Existence of Solutions for the Transport-Elliptic Equation
		27.2 Spatial Derivative Estimates for the Solution to the Transport- Elliptic Equation
		27.3 Material Derivative Estimates for the Transport-Elliptic Equation
		27.4 Cutting Off the Solution to the Transport-Elliptic Equation
Acknowledgments
Appendices
	A The Positive Direction of Onsager\'s Conjecture
	B Simplifications and Recent Developments
References
Index