Nonlinear Conservation Laws, Fluid Systems and Related Topics

Contents......Page 8
Preface......Page 6
1 Introduction......Page 10
2.1 Derivation of the Euler equations......Page 11
2.2.2 Vorticity-Stream function formulation......Page 15
2.3.1 Local conserved quantities......Page 17
2.3.2 Global conserved quantities......Page 19
2.4.1 Axisymmetric flow......Page 20
2.4.3 Jets and strains......Page 21
3 Local well-posedness of the 3D Euler equation......Page 22
3.1.1 Sobolev spaces......Page 23
3.1.2 Hodge decomposition and the Leray projection......Page 24
3.1.3 The Aubin-Lions lemma......Page 25
3.1.4 Calculus inequalities......Page 26
3.1.5 Gronwall\'s inequality......Page 27
3.2 Properties of mollifiers......Page 28
3.3 Global existence of the mollified equation......Page 30
3.4 Local existence of the Euler equations......Page 33
4.1 The Beale-Kato-Majda criterion......Page 35
4.2 Improvements of the BKM criterion......Page 38
5.1 Sufficient conditions by Constantin-FeffermanMajda......Page 40
5.2 Sufficient conditions by Deng-Hou-Yu......Page 47
6.1 1-D model......Page 53
6.2 The 2-D QG equation......Page 54
6.2.1 Existence and blow-up criteria......Page 56
6.2.2 Global existence result by Constantin-Majda-Tabak......Page 58
6.2.3 Global existence result by Cordoba and Fefferman......Page 62
6.2.4 Final remarks about the QG equation......Page 64
7.1 The contour dynamics equation (eDE)......Page 65
7.2 Levelset formulation and global existence......Page 66
References......Page 78
1 Hyperbolic systems of conservation laws......Page 81
1.1.1 Hyperbolicity......Page 82
1.1.2 Entropies......Page 83
1.1.3 Local well-posedness in HS(jRd)......Page 84
1.2.1 Break-down of smooth solutions......Page 85
1.2.2 Weak solutions......Page 86
1.2.3 The Rankine-Hugoniot condition......Page 87
1.2.4 Non-uniqueness of weak solutions......Page 88
1.2.5 Entropy admissibility condition......Page 89
1.2.6 The viscosity approach......Page 90
1.2.7 The scalar case......Page 91
1.3.1 The Hugoniot locus......Page 92
1.3.2 Genuine nonlinearity......Page 94
1.3.4 Viscous shock profiles......Page 95
1.4 The Riemann problem......Page 96
1.4.2 Contact discontinuities......Page 97
1.4.3 The theorem of Lax......Page 98
1.5 Existence of viscous shock profiles......Page 99
1.5.1 The scalar case......Page 100
1.5.2 Reduction to a center manifold (bifurcation analysis)......Page 101
1.5.3 Lax shocks......Page 103
1.5.4 Under-compressive shocks......Page 104
2 Finite difference schemes......Page 105
2.1 Conservative schemes......Page 106
2.1.2 Order of accuracy......Page 107
2.1.3 Linearized L2-stability......Page 110
2.1.4 The Courant-Friedrichs-Lewy condition......Page 112
2.1.5 Entropy-consistent schemes......Page 113
2.2.1 The naive centered scheme......Page 114
2.2.2 The Lax-Friedrichs scheme......Page 115
2.2.4 The Godunov scheme......Page 116
2.3 Schemes for scalar equations......Page 118
2.3.1 Monotone schemes......Page 119
2.3.2 Kutznetsov\'s error estimate......Page 120
3.1 DSPs and conservation......Page 121
3.1.1 The function Y......Page 123
3.1.2 Scalar case: monotone schemes......Page 124
3.2.1 DSPs for small steady Lax shocks......Page 126
3.2.2 DSPs for steady Lax shocks: the Godunov scheme......Page 129
3.2.3 What can go wrong?......Page 130
References......Page 132
1.1 Overview......Page 135
1.2.1 Intuitive derivation......Page 138
1.2.2 Collision invariants and H functional......Page 140
1.2.4 Basic properties of the collision operators......Page 147
2 Expansions and their unification......Page 156
2.1 Classical expansions......Page 157
2.2 Unification by decomposition......Page 163
3 Detour to hyperbolic conservation laws......Page 167
3.1 Scalar conservation laws......Page 168
3.2 Riemann problem for systems......Page 172
3.3.1 Existence......Page 176
3.3.2 Stability and uniqueness......Page 178
3.4 Vanishing viscosity......Page 184
4 Spectral analysis on the linearized Boltzmann operator......Page 186
4.1 Smoothing properties of etA......Page 187
4.2 Spectral properties of B......Page 191
4.3 Decay rates of etB in Xf3......Page 195
4.4 Effect of external force......Page 201
5.1 Global existence......Page 203
5.2 Optimal convergence rates......Page 208
5.3 External force, revisited......Page 220
References......Page 232
1.1 Why statistical description......Page 239
1.2 What characterizes statistical behavior......Page 240
2 Stationary statistics......Page 242
2.2 Definition, existence......Page 243
2.3 Ergodicity......Page 246
2.4 Invariant measure, stationary statistical solution and attractor......Page 258
2.5 Dependence on parameters......Page 260
2.6 Regular perturbation......Page 261
2.7 Singular perturbation......Page 265
2.8.1 An application to NSE: energy dissipation rate per unit mass......Page 271
2.8.2 An application to RBC: heat transfer in the vertical direction (N usselt number)......Page 273
2.9 Maximum entropy principle......Page 276
2.10 Application to ODEs......Page 283
2.11 Application to basic geophysical systems......Page 287
3.1 Definition, existence......Page 302
3.2.1 Reynolds equation for the average flow......Page 303
Appendix: some useful theorems......Page 304
References......Page 307
Introduction......Page 310
1.3 Mathematical treatment......Page 311
2 Characteristic decomposition of the pseudo-steady case......Page 314
2.1 Riemann problems......Page 315
2.2 Isentropic system......Page 317
2.3 Some explicit solutions......Page 318
2.4 A characteristic decomposition......Page 319
2.4.1 Introduction to the method of characteristics......Page 320
2.4.2 Decomposition......Page 323
3.1 Primary system......Page 327
3.2 The concept of hodograph transformation......Page 328
3.2.1 The hodograph transformation for the pseudo-steady Euler......Page 329
3.2.2 Steady Euler......Page 330
3.3 Characteristics in both planes......Page 331
3.4 Phase space system of equations......Page 332
3.5 Planar rarefaction waves......Page 337
3.6 The gas expansion problem......Page 338
3.6.2 A wedge of gas......Page 339
3.6.3 A wedge of gas in the hodograph plane......Page 341
3.6.4 Local existence......Page 343
3.6.5 Statement of main existence......Page 344
3.6.6 The maximum norm estimate on (o:,(3,c)......Page 345
3.6.7 Gradient estimates and the proof of Theorem 3.3......Page 348
3.6.8 Inversion......Page 350
3.6.9 Proof of Theorem 3.4......Page 352
3.6.10 Properties of the solutions......Page 353
3.7 Summary remarks......Page 356
3.8.1 Concept of simple waves......Page 357
3.8.2 Simple waves for pseudo-steady Euler equations......Page 358
Appendix B: convertibility......Page 360
4.1 Introduction......Page 361
4.2 Existence of solutions to the Cauchy problem......Page 363
4.2.1 Primary representations......Page 364
4.2.2 Primary estimates......Page 365
4.2.3 Estimates on modulus of continuity......Page 368
4.3 Goursat problem......Page 370
4.4 Mixed initial-boundary value problem......Page 376
4.5 Application to 2-D Euler......Page 377
5.1 Basic theorems......Page 378
6.1 Introduction......Page 382
6.1.1 Derivation......Page 383
6.1.2 Progress of research......Page 384
6.1.3 One-dimensional planar waves......Page 385
6.3 Subsonic region......Page 386
7 Open problems......Page 392
Epilogue: Stories......Page 395
References......Page 396