Mathematical Analysis of the Navier-Stokes Equations -- Cetraro, Italy 2017

Preface......Page 6
Contents......Page 7
1 Analysis of Viscous Fluid Flows: An Approach by Evolution Equations......Page 8
 1.1 Basics on Distributions, Fourier Transforms and Sobolev Spaces......Page 10
 1.2 The Vector Valued Setting......Page 18
 1.3 Cauchy Problems and Semigroups......Page 24
 1.4 Sectorial Operators and Bounded H∞-Functional Calculus......Page 32
 1.5 Fractional Powers......Page 40
 1.6 Operator-Valued H∞-Calculus, R-Boundedness, Fourier Multipliers and Maximal Lp-Regularity......Page 43
 1.7 Quasilinear Evolution Equations......Page 55
 1.8 Balance Laws......Page 61
 1.9 The Stokes Equation in a Half Space......Page 64
 1.10 The Stokes and Hydrostatic Stokes Equations on Domains......Page 78
 1.11 Nonlinear Stability of Ekman Boundary Layers......Page 88
 1.12 Fluid-Rigid Body Interaction Problems for Compressible Fluids......Page 100
 1.13 Two-Phase Free Boundary Value Problems for a Class of Non-Newtonian Fluids......Page 106
 1.14 Nematic Liquid Crystals......Page 118
 1.15 Global Strong Well-Posedness of the Primitive Equations......Page 127
 1.16 Justification of the Hydrostatic Approximation......Page 132
 1.17 Notes......Page 139
 References......Page 147
 2.1 Introduction......Page 154
  2.1.1 Notation and Preliminaries......Page 155
  2.2.1 Weak Solutions......Page 157
  2.2.2 Strong Solutions......Page 159
  2.2.3 Weak-Strong Uniqueness......Page 163
  2.2.4 Regular and Singular Times......Page 164
   2.2.5.1 Dimensions......Page 166
   2.2.5.2 Dimension of the Set of Singular Times......Page 168
 2.3 Serrin\'s Local Regularity Result for uL5+(Q*)......Page 170
  2.3.1 Step 1: Show that ωL∞(Qs*), s<1......Page 173
  2.3.2 Show that uC∞x(Qs*) for any s<1......Page 175
  2.3.3 The Case uL5(Q*)......Page 178
 2.4 Space-Time Partial Regularity......Page 179
  2.4.1 The Navier–Stokes Inequality......Page 180
  2.4.2 Partial Regularity......Page 181
  2.4.3 Idea of the Proof of Theorem 4.2......Page 183
  2.4.4 Interpolation Inequality......Page 184
  2.4.5 Inductive Argument......Page 185
 2.5 Lagrangian Trajectories......Page 189
  2.5.1 Volume-Preserving Solutions When u is Smooth......Page 190
  2.5.3 Existence of Trajectories When u is a Weak Solution......Page 191
  2.5.4 Existence of a Flow When u is a Weak Solution......Page 194
  2.5.5 Properties of the Flow Mapping......Page 195
 References......Page 197
  3.1.1 Free Boundary Problem for the Navier–Stokes Equations......Page 200
  3.1.2 Derivation of Boundary Conditions......Page 203
  3.1.3 Short History......Page 208
  3.1.4 Further Notation......Page 210
  3.2.1 Definitions of Domains......Page 212
  3.2.2 Besov Spaces on Γ......Page 219
  3.2.3 The Weak Dirichlet Problem......Page 220
  3.2.4 The Weak Laplace Problem in RN......Page 222
  3.2.5 The Weak Dirichlet Problem in the Half Space Case......Page 224
  3.2.6 Regularity of the Weak Dirichlet Problem......Page 227
  3.2.7 Laplace–Beltrami Operator......Page 232
  3.2.8 Parametrized Surface......Page 238
  3.2.9 Example of Mean Curvature......Page 240
  3.3.1 Hanzawa Transform......Page 244
  3.3.2 Transformation of Equations and the Divergence Free Condition......Page 246
  3.3.3 Transformation of the Boundary Conditions......Page 249
  3.3.4 Linearization Principle......Page 258
  3.4.1 Statement of Maximal Regularity Theorems......Page 260
  3.4.2 R Bounded Solution Operators......Page 267
  3.4.3 Stokes Operator and Reduced Stokes Operator......Page 271
  3.4.4 R-Bounded Solution Operators for the Reduced Stokes Equations......Page 274
  3.4.5 Generation of C0 Analytic Semigroup......Page 276
  3.4.6 Proof of Maximal Regularity Theorem......Page 281
  3.5.1 Model Problem in RN; Constant μ Case......Page 287
  3.5.2 Perturbed Problem in RN......Page 291
  3.5.3 Model Problem in RN+......Page 294
  3.5.4 Problem in a Bent Half Space......Page 321
  3.5.5 Some Preparation for the Proof of Theorem 3.4.11......Page 336
  3.5.6 Parametrix of Solutions of Eq.(3.184)......Page 338
  3.5.7 Estimates of the Remainder Terms......Page 343
  3.5.8 Proof of Theorem 3.4.11, Existence Part for Eq.(3.226)......Page 355
  3.5.9 A Proof of the Existence Part of Theorem 3.4.11......Page 356
  3.5.10 Uniqueness......Page 358
  3.5.11 A Priori Estimate......Page 360
 3.6 Local Well-Posedness for Arbitrary Initial Velocity Fields in a Uniform C3 Domain......Page 365
 3.7 Global Well-Posedness in a Bounded Domain Closed to a Ball......Page 380
  3.7.1 Derivation of Nonlinear Term hN(u, Ψρ) in Eq.(3.449)......Page 384
  3.7.2 Local Well-Posedness......Page 387
  3.7.3 Decay Estimates of Solutions for the Linearized Equations......Page 394
  3.7.4 Exponential Stability of Continuous Analytic Semigroup Associated with Eq.(3.489)......Page 403
  3.7.5 Global Wellposedness, a Proof of Theorem 3.7.1......Page 411
 3.8 Global Well-Posedness in an Exterior Domain......Page 419
  3.8.1 Maximal Lp–Lq Regularity in an Exterior Domain......Page 423
  3.8.2 Local Well-Posedness of Eq.(3.547)......Page 428
  3.8.3 A New Formulation of Eq.(3.547)......Page 435
  3.8.4 Slightly Perturbed Stokes Equations......Page 440
  3.8.5 Estimates for the Nonlinear Terms......Page 446
  3.8.6 A Proof of Theorem 3.8.1......Page 454
 References......Page 466
Lecture Notes in Mathematics......Page 470