Singular Limits in Thermodynamics of Viscous Fluids (eBook)

Contents 6
Preface 12
Notation, Definitions, and Function Spaces 17
0.1 Notation 17
0.2 Differential operators 19
0.3 Function spaces 20
0.4 Sobolev spaces 25
0.5 Fourier transform 30
0.6 Weak convergence of integrable functions 33
0.7 Non-negative Borel measures 34
0.8 Parametrized (Young) measures 35
Fluid Flow Modeling 37
1.1 Fluids in continuum mechanics 38
1.2 Balance laws 40
1.3 Field equations 44
1.4 Constitutive relations 49
Weak Solutions, A Priori Estimates 54
2.1 Weak formulation 56
2.2 A priori estimates 60
Existence Theory 77
3.1 Hypotheses 78
3.2 Structural properties of constitutive functions 81
3.3 Main existence result 84
3.4 Solvability of the approximate system 87
3.5 Faedo-Galerkin limit 103
3.6 Artificial diffusion limit 119
3.7 Vanishing artificial pressure 138
3.8 Regularity properties of the weak solutions 156
Asymptotic Analysis – An Introduction 161
4.1 Scaling and scaled equations 163
4.2 Low Mach number limits 165
4.3 Strongly stratified flows 167
4.4 Acoustic waves 169
4.5 Acoustic analogies 173
4.6 Initial data 175
4.7 A general approach to singular limits for the full Navier- Stokes- Fourier system 176
Singular Limits – Low Stratification 180
5.1 Hypotheses and global existence for the primitive system 183
5.2 Dissipation equation, uniform estimates 186
5.3 Convergence 193
5.4 Convergence of the convective term 202
5.5 Conclusion – main result 216
Stratified Fluids 227
6.1 Motivation 227
6.2 Primitive system 228
6.3 Asymptotic limit 233
6.4 Uniform estimates 238
6.5 Convergence towards the target system 246
6.6 Analysis of acoustic waves 252
6.7 Asymptotic limit in entropy balance 260
Interaction of Acoustic Waves with Boundary 263
7.1 Problem formulation 265
7.2 Main result 268
7.3 Uniform estimates 271
7.4 Analysis of acoustic waves 273
7.5 Strong convergence of the velocity field 285
Problems on Large Domains 293
8.1 Primitive system 293
8.2 Uniform estimates 296
8.3 Acoustic equation 300
8.4 Regularization and extension to 303
8.5 Dispersive estimates and time decay of the acoustic waves 309
8.6 Conclusion – main result 314
Acoustic Analogies 316
9.1 Asymptotic analysis and the limit system 317
9.2 Acoustic equation revisited 318
9.3 Two-scale convergence 322
9.4 Lighthill’s acoustic analogy in the low Mach number regime 327
9.5 Concluding remarks 331
Appendix 333
10.1 Mollifiers 333
10.2 Basic properties of some elliptic operators 334
10.3 Normal traces 341
10.4 Singular and weakly singular operators 344
10.5 The inverse of the div-operator ( Bogovskii’s formula) 345
10.6 Helmholtz decomposition 353
10.7 Function spaces of hydrodynamics 355
10.8 Poincar ´ e type inequalities 357
10.9 Korn type inequalities 359
10.10 Estimating 363
u by means of 363
and curlxu 363
10.11 Weak convergence and monotone functions 364
10.12 Weak convergence and convex functions 368
10.13 Div-Curl lemma 371
10.14 Maximal regularity for parabolic equations 373
10.15 Quasilinear parabolic equations 375
10.16 Basic properties of the Riesz transform and related operators 377
10.17 Commutators involving Riesz operators 380
10.18 Renormalized solutions to the equation of continuity 382
Bibliographical Remarks 389
11.1 Fluid flow modeling 389
11.2 Mathematical theory of weak solutions 390
11.3 Existence theory 391
11.4 Analysis of singular limits 391
11.5 Propagation of acoustic waves 392
Bibliography 393
Index 406