Hölder continuous Euler flows in three dimensions with compact support in time:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Princeton ; Oxford
Princeton University Press
2017
|
| Schriftenreihe: | Annals of mathematics studies
Number 196 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | x, 201 Seiten |
| ISBN: | 9780691174822 9780691174839 |
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Datensatz im Suchindex
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| adam_text | Titel: Hölder continuous Euler flows in three dimensions with compact support in time
Autor: Isett, Philip
Jahr: 2017
Contents Preface ix I Introduction 1 1 The Euler-Reynolds System 7 II General Considerations of the Scheme 11 2 Structure of the Book 13 3 Basic Technical Outline 14 III Basic Construction of the Correction 19 4 Notation 19 5 A Main Lemma for Continuous Solutions 20 6 The Divergence Equation 22 6.1 A Remark about Momentum Conservation 22 6.2 The Parametrix 24 6.3 Higher Order Parametrix Expansion 27 6.4 An Inverse for Divergence 28 7 Constructing the Correction 30 7.1 Transportation of the Phase Functions 30 7.2 The High-High Interference Problem and Beltrami Flows 32 7.3 Eliminating the Stress 36 7.3.1 The Approximate Stress Equation 36 7.3.2 The Stress Equation and the Initial Phase Directions 38 7.3.3 The Index Set, the Cutoffs and the Phase Functions 40 7.3.4 Localizing the Stress Equation 45 7.3.5 Solving the Quadratic Equation 46 7.3.6 The Renormalized Stress Equation in Scalar Form 50 7.3.7 Summary 54 IV Obtaining Solutions from the Construction 56
CONTENTS 8 Constructing Continuous Solutions 56 8.1 Step 1: Mollifying the Velocity 56 8.2 Step 2: Mollifying the Stress 57 8.3 Step 3: Choosing the Lifespan 58 8.4 Step 4: Bounds for the New Stress 59 8.5 Step 5: Bounds for the Corrections 60 8.6 Step 6: Control of the Energy Increment 60 9 Frequency and Energy Levels 62 10 The Main Iteration Lemma 67 10.1 Frequency Energy Levels for the Euler-Reynolds Equations 67 10.2 Statement of the Main Lemma 68 11 Main Lemma Implies the Main Theorem 71 11.1 The Base Case 72 11.2 The Main Lemma Implies the Main Theorem 75 11.2.1 Choosing the Parameters 75 11.2.2 Choosing the Energies 77 11.2.3 Regularity of the Velocity Field 78 11.2.4 Asymptotics for the Parameters 81 11.2.5 Regularity of the Pressure 85 11.2.6 Compact Support in Time 86 11.2.7 Nontriviality of the Solution 87 12 Gluing Solutions 89 13 On Onsager’s Conjecture 90 13.1 Higher Regularity for the Energy 92 V Construction of Regular Weak Solutions: Preliminaries 97 14 Preparatory Lemmas 97 15 The Coarse Scale Velocity 99 16 The Coarse Scale Flow and Commutator Estimates 102 17 Transport Estimates 105 17.1 Stability of the Phase Functions 106 17.2 Relative Velocity Estimates 108 17.3 Relative Acceleration Estimates 113 vi
CONTENTS 18 Mollification along the Coarse Scale Flow 115 18.1 The Problem of Mollifying the Stress in Time 115 18.2 Mollifying the Stress in Space and Time 116 18.3 Choosing Mollification Parameters 116 18.4 Estimates for the Coarse Scale Flow 119 18.5 Spatial Variations of the Mollified Stress 122 18.6 Transport Estimates for the Mollified Stress 123 18.6.1 Derivatives and Averages along the Flow Commute 124 18.6.2 Material Derivative Bounds for the Mollified Stress 126 18.6.3 Second Time Derivative of the Mollified Stress along the Coarse Scale Flow 129 18.6.4 An Acceptability Check 130 19 Accounting for the Parameters and the Problem with the High-High Term 131 VI Construction of Regular Weak Solutions: Estimating the Correction 135 20 Bounds for Coefficients from the Stress Equation 135 21 Bounds for the Vector Amplitudes 138 22 Bounds for the Corrections 143 22.1 Bounds for the Velocity Correction 143 22.2 Bounds for the Pressure Correction 146 23 Energy Approximation 147 24 Checking Frequency Energy Levels for the Velocity and Pressure 150 VII Construction of Regular Weak Solutions: Estimating the New Stress 152 25 Stress Terms Not Involving Solving the Divergence Equation 155 25.1 The Mollification Term from the Velocity 156 25.2 The Mollification Term from the Stress 161 25.3 Estimates for the Stress Term 162 26 Terms Involving the Divergence Equation 163 26.1 Expanding the Parametrix 164 26.2 Applying the Parametrix 168 vii
CONTENTS 27 Transport-Elliptic Estimates 173 27.1 Existence of Solutions for the Transport-Elliptic Equation 176 27.2 Spatial Derivative Estimates for the Solution to the Transport- Elliptic Equation 179 27.3 Material Derivative Estimates for the Transport-Elliptic Equation 181 27.4 Cutting Off the Solution to the Transport-Elliptic Equation 182 Acknowledgments 183 Appendices 185 A The Positive Direction of Onsager’s Conjecture 191 B Simplifications and Recent Developments 194 References 197 Index 201 viii
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| author | Isett, Philip 1986- |
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| dewey-full | 532/.05 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
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| discipline | Physik Mathematik |
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| spelling | Isett, Philip 1986- Verfasser (DE-588)1128680203 aut Hölder continuous Euler flows in three dimensions with compact support in time Philip Isett Princeton ; Oxford Princeton University Press 2017 x, 201 Seiten txt rdacontent n rdamedia nc rdacarrier Annals of mathematics studies Number 196 Includes bibliographical references and index Fluid dynamicsxMathematics Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Strömungsmechanik (DE-588)4077970-1 gnd rswk-swf Eulersche Formel (DE-588)4359957-6 gnd rswk-swf Hölder-Stetigkeit (DE-588)4332993-7 gnd rswk-swf Eulersche Bewegungsgleichungen (DE-588)4219070-8 gnd rswk-swf 1\p (DE-588)4113937-9 Hochschulschrift gnd-content Eulersche Bewegungsgleichungen (DE-588)4219070-8 s Hölder-Stetigkeit (DE-588)4332993-7 s Strömungsmechanik (DE-588)4077970-1 s DE-604 Partielle Differentialgleichung (DE-588)4044779-0 s Eulersche Formel (DE-588)4359957-6 s 2\p DE-604 Annals of mathematics studies Number 196 (DE-604)BV000000991 196 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029643012&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Isett, Philip 1986- Hölder continuous Euler flows in three dimensions with compact support in time Annals of mathematics studies Fluid dynamicsxMathematics Partielle Differentialgleichung (DE-588)4044779-0 gnd Strömungsmechanik (DE-588)4077970-1 gnd Eulersche Formel (DE-588)4359957-6 gnd Hölder-Stetigkeit (DE-588)4332993-7 gnd Eulersche Bewegungsgleichungen (DE-588)4219070-8 gnd |
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| title | Hölder continuous Euler flows in three dimensions with compact support in time |
| title_auth | Hölder continuous Euler flows in three dimensions with compact support in time |
| title_exact_search | Hölder continuous Euler flows in three dimensions with compact support in time |
| title_full | Hölder continuous Euler flows in three dimensions with compact support in time Philip Isett |
| title_fullStr | Hölder continuous Euler flows in three dimensions with compact support in time Philip Isett |
| title_full_unstemmed | Hölder continuous Euler flows in three dimensions with compact support in time Philip Isett |
| title_short | Hölder continuous Euler flows in three dimensions with compact support in time |
| title_sort | holder continuous euler flows in three dimensions with compact support in time |
| topic | Fluid dynamicsxMathematics Partielle Differentialgleichung (DE-588)4044779-0 gnd Strömungsmechanik (DE-588)4077970-1 gnd Eulersche Formel (DE-588)4359957-6 gnd Hölder-Stetigkeit (DE-588)4332993-7 gnd Eulersche Bewegungsgleichungen (DE-588)4219070-8 gnd |
| topic_facet | Fluid dynamicsxMathematics Partielle Differentialgleichung Strömungsmechanik Eulersche Formel Hölder-Stetigkeit Eulersche Bewegungsgleichungen Hochschulschrift |
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