Hölder Continuous Euler Flows In Three Dimensions With Compact Support In Time

Synopsis

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. Philip Isett. Author's Thesis (doctoral)--princeton University, Princeton, N.j., 2013. Includes Bibliographical References And Index.