Preface ix
Notation xi
Chapter 1. Introduction 1
§1.1. Navier-Stokes equations 1
§1.2. Derivation of Navier-Stokes equations 3
§1.3. Scaling and a priori estimates 6
§1.4. Vorticity 7
§1.5. Pressure 10
§1.6. Helmholtz decomposition 13
§1.7. Notes 17
Problems 17
Chapter 2. Steady states 19
§2.1. Weak solutions 19
§2.2. Small-large uniqueness 22
§2.3. Existence for zero boundary data by the Galerkin method 23
§2.4. Existence for zero boundary data by the Leray-Schauder
theorem 25
§2.5. Nonuniqueness 29
§2.6. L q -theory for the linear system 32
§2.7. Regularity 38
§2.8. The Bogovskii map 45
§2.9. Notes 47
Problems 48
v
vi Contents
Chapter 3. Weak solutions 51
§3.1. Weak form, energy inequalities, and definitions 51
§3.2. Auxiliary results 55
§3.3. Existence for the perturbed Stokes system 58
§3.4. Compactness lemma 60
§3.5. Existence of suitable weak solutions 62
§3.6. Notes 67
Problems 68
Chapter 4. Strong solutions 69
§4.1. Dimension analysis 70
§4.2. Uniqueness 71
§4.3. Regularity 75
§4.4. Notes 77
Problems 77
Chapter 5. Mild solutions 79
§5.1. Nonstationary Stokes system and Stokes semigroup 79
§5.2. Existence of mild solutions 83
§5.3. Applications to weak solutions 89
§5.4. Notes 92
Problems 92
Chapter 6. Partial regularity 93
§6.1. The set of singular times 94
§6.2. The set of singular space-time points 96
§6.3. Regularity criteria in scaled norm 97
§6.4. Notes 105
Problems 106
Chapter 7. Boundary value problem and bifurcation 107
§7.1. Existence: A priori bound by a good extension 108
§7.2. Existence: A priori bound by contradiction 112
§7.3. The Korobkov-Pileckas-Russo approach for 2D BVP 116
§7.4. The bifurcation problem and degree 123
§7.5. Bifurcation of the Rayleigh-Benard convection 128
§7.6. Bifurcation of Couette-Taylor flows 133
§7.7. Notes 139
Problems 140
Contents vii
Chapter 8. Self-similar solutions 141
§8.1. Self-similar solutions and similarity transform 141
§8.2. Stationary self-similar solutions 145
§8.3. Backward self-similar solutions 150
§8.4. Forward self-similar solutions 158
§8.5. Notes 171
Problems 171
Chapter 9. The uniform L 3 class 173
§9.1. Uniqueness 174
§9.2. Auxiliary results for regularity 176
§9.3. Regularity 178
§9.4. Backward uniqueness and unique continuation 184
§9.5. Notes 187
Chapter 10. Axisymmetric flows 189
§10.1. Axisymmetric Navier-Stokes equations 189
§10.2. No swirl case 195
§10.3. Type I singularity: De Giorgi-Nash-Moser approach 197
§10.4. Type I singularity: Liouville theorem approach 206
§10.5. Connections between the two approaches 209
§10.6. Notes 210
Bibliography 211
Index 223
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