The Mathematical Analysis of the Incompressible Euler and Navier-stokes Equations: An Introduction

235 Pages·2022·6.692 MB·English

by  Jacob Bedrossian, Vlad Vicol

225
GRADUATE STUDIES
IN MATHEMATICS
The Mathematical
Analysis of the
Incompressible Euler
and Navier-Stokes
Equations
An Introduction
Jacob Bedrossian
Vlad Vicol
The Mathematical
Analysis of the
Incompressible Euler
and Navier-Stokes
Equations
An Introduction

225
GRADUATE STUDIES
IN MATHEMATICS
The Mathematical
Analysis of the
Incompressible Euler
and Navier-Stokes
Equations
An Introduction
Jacob Bedrossian
Vlad Vicol
EDITORIAL COMMITTEE
Matthew Baker
Marco Gualtieri
Gigliola Staffilani (Chair)
Jeff A. Viaclovsky
Rachel Ward
2020 Mathematics Subject Classification. Primary 35Q30, 35Q31, 35Q35, 76Bxx,
76Dxx, 76E05.
For additional informationand updates on this book, visit
www.ams.org/bookpages/gsm-225
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DOI:https://doi.org/10.1090/gsm/225
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Contents
Preface xi
Chapter 1. Ideal Incompressible Fluids: The Euler Equations 1
§1.1. Eulerian vs Lagrangian representations 1
§1.2. Incompressibility and transport 4
§1.3. The incompressible homogeneous Euler equations 7
§1.4. Vorticity 15
§1.5. Symmetries and conservation laws 24
§1.6. Special explicit solutions 26
§1.7. Exercises 31
Chapter 2. Existence of Solutions and Continuation Criteria for Euler 33
§2.1. Local existence of Hs solutions 33
§2.2. The Lipschitz continuation criterion 45
§2.3. The Beale-Kato-Majda theorem 47
§2.4. The global existence of strong solutions in 2D 51
§2.5. The Constantin-Fefferman-Majda criterion 51
§2.6. Exercises 54
Chapter 3. Incompressible Viscous Fluids: The Navier-Stokes
Equations 59
§3.1. Viscosity 59
§3.2. Nondimensionalization 61
§3.3. Vorticity, symmetries, and balance laws 63
vii
viii Contents
§3.4. Special explicit solutions 65
§3.5. Local existence of Hs solutions 67
§3.6. Strong solutions with initial data in H1: Local and global 74
§3.7. Exercises 78
Chapter 4. Leray-Hopf Weak Solutions of Navier-Stokes 81
§4.1. Weak solutions 81
§4.2. Existenceofweaksolutionsonthewholespaceviamollification 84
§4.3. The uniqueness of weak solutions in 2D 89
§4.4. Weak-strong uniqueness and the Prodi-Serrin class 90
§4.5. Partial regularity in time for Leray-Hopf weak solutions 95
§4.6. Existence of weak solutions on the periodic box via Galerkin 97
§4.7. Exercises 102
Chapter 5. Mild Solutions of Navier-Stokes 105
§5.1. Mild formulation 105
§5.2. Scaling criticality 106
§5.3. Local-in-time well-posedness in H˙1/2 108
§5.4. Local-in-time well-posedness in L3 114
§5.5. Local regularization 120
§5.6. Continuation of smooth solutions 121
§5.7. Exercises 124
Chapter 6. A Survey of Some Advanced Topics 127
§6.1. Local regularity and the Prodi-Serrin conditions 127
§6.2. Partial regularity of suitable weak solutions in 3D 129
§6.3. Bounded domains 135
§6.4. Stationary solutions of the Navier-Stokes equations 144
§6.5. Ruling out backward self-similar finite-time singularities 145
§6.6. Critical and supercritical well-posedness for Navier-Stokes 147
§6.7. Yudovich theory and 2D Euler with Lp vorticity 150
§6.8. Gradient growth in the 2D Euler equations 151
§6.9. The search for finite-time singularties in 3D Euler 153
§6.10. Hydrodynamic stability: Euler 154
§6.11. Hydrodynamic stability: Navier-Stokes 158
§6.12. The energy balance and Onsager’s conjecture 161
Contents ix
Appendix 169
§A.1. The contraction mapping principle 169
§A.2. Existence and uniqueness for ODEs 170
§A.3. Fourier transform 171
§A.4. Integral operators 172
§A.5. Sobolev spaces 176
§A.6. Basic properties of the Poisson and heat equations 179
§A.7. Mollifiers 181
§A.8. Sobolev and Gagliardo-Nirenberg inequalities 183
§A.9. Compactness 188
Bibliography 199
Index 217