Preface ......................................................... v
I The equations of fluid mechanics ............................. 1
1 Continuous description of a fluid ............................ 1
1.1 The continuous medium assumption. Density ............... 1
1.2 Lagrangian and Eulerian coordinates ..................... 3
2 The transport theorem ........................................ 5
3 Evolution equations .......................................... 7
3.1 Balance equations ....................................... 8
3.2 Cauchy's stress theorem ................................ 11
3.3 Evolution equations revisited .......................... 17
4 Fundamental laws: Newtonian fluids and thermodynamics laws .. 19
4.1 Fluids at rest ......................................... 20
4.2 Newton's hypothesis .................................... 20
4.3 Consequences of the second law of thermodynamics ....... 25
4.4 Equation for the specific internal energy .............. 28
4.5 Formulation in entropy and temperature ................. 29
5 Summary of the equations .................................... 30
6 Incompressible models ....................................... 32
6.1 The incompressibility assumption ....................... 32
6.2 Overview of the incompressible models .................. 37
7 Some exact steady solutions ................................. 42
7.1 Poiseuille flow in a pipe .............................. 43
7.2 Planar shear flow ...................................... 44
7.3 Couette flow between two cylinders ..................... 46
II Analysis tools ............................................. 49
1 Main notation ............................................... 50
2 Fundamental results from functional analysis ................ 51
2.1 Banach spaces .......................................... 51
2.2 Weak and weak-* convergences ........................... 52
2.3 Lebesgue spaces ........................................ 56
2.4 Partitions of unity .................................... 68
2.5 A short introduction to distribution theory ............ 71
2.6 Lipschitz continuous functions ......................... 75
3 Basic compactness results ................................... 77
3.1 Compact sets in function spaces ........................ 77
3.2 Compact maps ........................................... 79
3.3 The Schauder fixed-point theorem ....................... 83
4 Functions of one real variable .............................. 84
4.1 Differentiation and antiderivatives .................... 84
4.2 Differential inequalities and Gronwall's lemma ......... 88
5 Spaces of Banach-valued functions ........................... 92
5.1 Definitions and main properties ........................ 92
5.2 Regularity in time ..................................... 94
5.3 Compactness theorems .................................. 102
5.4 Banach-valued Fourier transform ....................... 106
6 Some results in spectral analysis of unbounded operators ... 110
6.1 Definitions ........................................... 110
6.2 Elementary results of spectral theory ................. 112
6.3 Applications to the semigroup theory .................. 118
III Sobolev spaces ............................................ 121
1 Domains .................................................... 122
1.1 General definitions ................................... 122
1.2 Lipschitz domains ..................................... 123
2 Sobolev spaces on Lipschitz domains ........................ 135
2.1 Definitions ........................................... 136
2.2 Mollifying operators and Friedrichs commutator
estimates ............................................. 138
2.3 Change of variables ................................... 149
2.4 Extension operator .................................... 150
2.5 Trace and trace lifting operators ..................... 153
2.6 Duality theory for Sobolev spaces ..................... 159
2.7 Translation estimates ................................. 164
2.8 Sobolev embeddings .................................... 167
2.9 Poincare and Hardy inequalities ....................... 179
2.10 Domains of first-order differential operators ......... 184
3 Calculus near the boundary of domains ...................... 189
3.1 Local charts description of the boundary .............. 189
3.2 Distance to the boundary. Projection on the boundary .. 191
3.3 Regularised distance .................................. 194
3.4 Parametrisation of a neighborhood of dQ. .............. 200
3.5 Tangential Sobolev spaces ............................. 206
3.6 Differential operators in tangential/normal
coordinates ........................................... 217
4 The Laplace problem ........................................ 222
4.1 Dirichlet boundary conditions ......................... 222
4.2 Neumann boundary conditions ........................... 226
IV Steady Stokes equations .................................... 229
1 Necas inequality ........................................... 230
1.1 Proof of the inequality ............................... 231
1.2 Related Poincare inequalities ......................... 238
2 Characterisation of gradient fields. De Rham's theorem ..... 241
3 The divergence operator and related spaces ................. 245
3.1 Right-inverse for the divergence ...................... 245
3.2 The space Hdw(Sl) ..................................... 248
3.3 Divergence-free vector fields. Leray decomposition .... 249
4 The curl operator and related spaces ....................... 252
4.1 Poincare's theorems ................................... 252
4.2 The space Hcml{Sl) .................................... 257
4.3 Kernel and image of the curl operator ................. 267
4.4 The div/curl problem .................................. 269
5 The Stokes problem ......................................... 273
5.1 Well-posedness of the Stokes problem .................. 273
5.2 Stokes operator ....................................... 277
5.3 The unsteady Stokes problem ........................... 286
5.4 Penalty approximation of the Stokes problem ........... 287
6 Regularity of the Stokes problem ........................... 290
6.1 First degree of regularity ............................ 290
6.2 Higher-order regularity ............................... 301
6.3 Lq theory of the Stokes problem ....................... 302
6.4 Regularity for the div/curl problem ................... 304
7 The Stokes problem with stress boundary conditions ......... 306
7.1 The Stokes-Neumann problem ............................ 307
7.2 Regularity properties ................................. 311
7.3 Stress boundary conditions ............................ 315
8 The interface Stokes problem ............................... 323
8.1 Existence and uniqueness .............................. 324
8.2 Regularity of the solution ............................ 326
9 The Stokes problem with vorticity boundary conditions ...... 329
9.1 Preliminaries ......................................... 330
9.2 A vector Laplace problem .............................. 332
9.3 The Stokes problem .................................... 340
V Navier-Stokes equations for homogeneous fluids ............. 345
1 Leray's theorem ............................................ 346
1.1 Properties of the inertia term ........................ 346
1.2 Weak formulations of the Navier-Stokes equations ...... 347
1.3 Existence and uniqueness of weak solutions ............ 352
1.4 Kinetic energy evolution .............................. 363
1.5 Existence and regularity of the pressure .............. 368
2 Strong solutions ........................................... 370
2.1 New estimates ......................................... 371
2.2 The two-dimensional case .............................. 373
2.3 The three-dimensional case ............................ 376
2.4 Parabolic regularity properties ....................... 384
2.5 Regularisation over time .............................. 389
3 The steady Navier-Stokes equations ......................... 391
3.1 The case of homogeneous boundary conditions ........... 392
3.2 The case of nonhomogeneous boundary conditions ........ 395
3.3 Uniqueness for small data ............................. 401
3.4 Asymptotic stability of steady solutions .............. 402
VI Nonhomogeneous fluids ..................................... 409
1 Weak solutions of the transport equation ................... 411
1.1 Setting of the problem ................................ 412
1.2 Trace theorem. Renormalisation property ............... 413
1.3 The initial- and boundary-value problem ............... 423
1.4 Stability theorem ..................................... 427
2 The nonhomogeneous incompressible Navier-Stokes equations .. 434
2.1 Main result ........................................... 434
2.2 Approximate problem ................................... 435
2.3 Estimates for the approximate solution ................ 443
2.4 End of the proof of the existence theorem ............. 447
2.5 The case without vacuum ............................... 452
VII Boundary conditions modelling ............................. 453
1 Outflow boundary conditions ................................ 454
1.1 Setting up the model .................................. 454
1.2 Existence and uniqueness .............................. 458
2 Dirichlet boundary conditions through a penalty method ..... 470
2.1 A simple example of a boundary layer .................. 472
2.2 Statement of the main result .......................... 476
2.3 Formal asymptotic expansion ........................... 479
2.4 Well-posedness of profile equations ................... 492
2.5 Convergence of the asymptotic expansion ............... 497
A Classic differential operators ............................. 507
1 The scalar and vector cases ............................. 507
1.1 Definitions ........................................ 507
1.2 Useful formulas .................................... 509
2 Extension to second-order tensors ....................... 509
В Thermodynamics supplement .................................. 511
1 Heat capacity ........................................... 512
2 The first law of thermodynamics. Internal energy ........ 512
3 The second law of thermodynamics ........................ 513
3.1 Entropy ............................................ 513
3.2 Internal energy calculation ........................ 514
4 Specific variables ...................................... 515
References ................................................. 517
Index ......................................................... 523
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