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Hölder continuous Euler flows in three dimensions with compact support in time

Author: Philip Isett
Publisher: Princeton : Princeton University Press, 2017. ©2017
Dissertation: Author's thesis (doctoral)--Princeton University, Princeton, N.J., 2013.
Series: Annals of mathematics studies, no. 196.
Edition/Format:   Thesis/dissertation : Thesis/dissertation : EnglishView all editions and formats
Publication:Annals of mathematics studies, no:196
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Material Type: Thesis/dissertation
Document Type: Book
All Authors / Contributors: Philip Isett
ISBN: 9780691174822 0691174822 9780691174839 0691174830
OCLC Number: 959080783
Description: x, 201 pages ; 25 cm.
Contents: Introduction --
The Euler-Reynolds System --
General Considerations of the Scheme --
Structure of the Book --
Basic Technical Outline --
Basic Construction of the Correction --
Notation --
A Main Lemma for Continuous Solutions --
The Divergence Equation --
A Remark about Momentum Conservation --
The Parametrix --
Higher Order Parametrix Expansion --
An Inverse for Divergence --
Constructing the Correction --
Transportation of the Phase Functions --
The High-High Interference Problem and Beltrami Flows --
Eliminating the Stress --
The Approximate Stress Equation --
The Stress Equation and the Initial Phase Directions --
The Index Set, the Cutoffs and the Phase Functions --
Localizing the Stress Equation --
Solving the Quadratic Equation --
The Renormalized Stress Equation in Scalar Form --
Summary --
Obtaining Solutions from the Construction --
Constructing Continuous Solutions --
Step 1: Mollifying the Velocity --
Step 2: Mollifying the Stress --
Step 3: Choosing the Lifespan --
Step 4: Bounds for the New Stress --
Step 5: Bounds for the Corrections --
Step 6: Control of the Energy Increment --
Frequency and Energy Levels --
The Main Iteration Lemma --
Frequency Energy Levels for the Euler-Reynolds Equations --
Statement of the Main Lemma --
Main Lemma Implies the Main Theorem --
The Base Case --
The Main Lemma Implies the Main Theorem --
Choosing the Parameters --
Choosing the Energies --
Regularity of the Velocity Field --
Asymptotics for the Parameters --
Regularity of the Pressure --
Compact Support in Time --
Nontriviality of the Solution --
Gluing Solutions --
On Onsager's Conjecture --
Higher Regularity for the Energy --
Construction of Regular Weak Solutions: Preliminaries --
Preparatory Lemmas --
The Coarse Scale Velocity --
The Coarse Scale Flow and Commutator Estimates --
Transport Estimates --
Stability of the Phase Functions --
Relative Velocity Estimates --
Relative Acceleration Estimates --
Mollification along the Coarse Scale Flow --
The Problem of Mollifying the Stress in Time --
Mollifying the Stress in Space and Time --
Choosing Mollification Parameters --
Estimates for the Coarse Scale Flow --
Spatial Variations of the Mollified Stress --
Transport Estimates for the Mollified Stress --
Derivatives and Averages along the Flow Commute --
Material Derivative Bounds for the Mollified Stress --
Second Time Derivative of the Mollified Stress along the Coarse Scale Flow --
An Acceptability Check --
Accounting for the Parameters and the Problem with the High-High Term --
Construction of Regular Weak Solutions: Estimating the Correction --
Bounds for Coefficients from the Stress Equation --
Bounds for the Vector Amplitudes --
Bounds for the Corrections --
Bounds for the Velocity Corrections --
Bounds for the Pressure Correction --
Energy Approximation --
Checking Frequency Energy Levels for the Velocity and Pressure --
Construction of Regular Weak Solutions: Estimating the New Stress --
Stress Terms Not Involving Solving the Divergence Equation --
The Mollification Term from the Velocity --
The Mollification Term from the Stress --
Estimates for the Stress Term --
Terms Involving the Divergence Equation --
Expanding the Parametrix --
Applying the Parametrix --
Transport-Elliptic Estimates --
Existence of Solutions for the Transport-Elliptic Equation --
Spatial Derivative Estimates for the Solution to the Transport-Elliptic Equation --
Material Derivative Estimates for the Transport-Elliptic Equation --
Cutting Off the Solution to the Transport-Elliptic Equation.
Series Title: Annals of mathematics studies, no. 196.
Responsibility: Philip Isett.

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