Many interesting problems in mathematical fluid dynamics involve the behavior of solutions of nonlinear systems of partial differential equations as certain parameters vanish or become infinite. Frequently the limiting solution, provided the limit exists, satisfies a qualitatively different system of differential equations. This book is designed as an introduction to the problems involving singular limits based on the concept of weak or variational solutions. The primitive system consists of a complete system of partial differential equations describing the time evolution of the three basic state variables: the density, the velocity, and the absolute temperature associated to a fluid, which is supposed to be compressible, viscous, and heat conducting. It can be represented by the Navier-Stokes-Fourier-system that combines Newton's rheological law for the viscous stress and Fourier's law of heat conduction for the internal energy flux. As a summary, this book studies singular limits of weak solutions to the system governing the flow of thermally conducting compressible viscous fluids.
Author(s): Eduard Feireisl, Antonin Novotny
Edition: 1
Year: 2009
Language: English
Pages: 418
Tags: Механика;Механика жидкостей и газов;
Cover......Page 1
Advances in Mathematical Fluid Mechanics......Page 3
Singular Limits in Thermodynamics of Viscous Fluids......Page 4
9783764388423......Page 5
Contents......Page 6
Preface......Page 12
Acknowledgement......Page 16
0.1 Notation......Page 18
0.2 Differential operators......Page 20
0.3 Function spaces......Page 21
0.4 Sobolev spaces......Page 26
0.5 Fourier transform......Page 31
0.6 Weak convergence of integrable functions......Page 34
0.7 Non-negative Borel measures......Page 35
0.8 Parametrized (Young) measures......Page 36
1 Fluid Flow Modeling......Page 38
1.1 Fluids in continuum mechanics......Page 39
1.2 Balance laws ......Page 41
1.3.1 Conservation of mass......Page 45
1.3.2 Balance of linear momentum ......Page 46
1.3.3 Total energy ......Page 48
1.3.4 Entropy ......Page 49
1.4.1 Molecular energy and transport terms ......Page 50
1.4.2 State equations ......Page 51
1.4.3 Effect of thermal radiation ......Page 53
1.4.4 Typical values of some physical coefficients ......Page 54
2 Weak Solutions, A Priori Estimates......Page 56
2.1 Weak formulation ......Page 58
2.1.2 Balance of linear momentum ......Page 59
2.1.3 Balance of total energy ......Page 60
2.1.5 Constitutive relations ......Page 61
2.2.1 Total mass conservation......Page 62
2.2.2 Energy estimates ......Page 63
2.2.3 Estimates based on the Second law of thermodynamics ......Page 64
2.2.4 Positivity of the absolute temperature ......Page 69
2.2.5 Pressure estimates ......Page 72
2.2.6 Pressure estimates, an alternative approach ......Page 76
3 Existence Theory......Page 80
3.1 Hypotheses ......Page 81
3.2 Structural properties of constitutive functions ......Page 84
3.3 Main existence result ......Page 87
3.3.1 Approximation scheme ......Page 88
3.4.1 Approximate continuity equation ......Page 90
3.4.2 Approximate internal energy equation ......Page 92
3.4.3 Local solvability of the approximate problem ......Page 99
3.4.4 Uniform estimates and global existence ......Page 101
3.5.1 Estimates independent of the dimension of Faedo-Galerkin approximations ......Page 106
3.5.2 Limit passage in the approximate continuity equation ......Page 109
3.5.3 Strong convergence of the approximate temperatures and the limit in the entropy equation ......Page 112
3.5.5 The limit system resulting from the Faedo-Galerkin approximation ......Page 119
3.5.6 The entropy production rate represented by a positive measure ......Page 121
3.6.1 Uniform estimates and limit in the approximate continuity equation ......Page 122
3.6.2 Entropy balance and strong convergence of the approximate temperatures ......Page 124
3.6.3 Uniform pressure estimates......Page 129
3.6.4 Limit in the approximate momentum equation and in the energy balance ......Page 131
3.6.5 Strong convergence of the densities ......Page 132
3.6.6 Artificial diffusion asymptotic limit ......Page 139
3.7.1 Uniform estimates ......Page 141
3.7.2 Asymptotic limit for vanishing artificial pressure ......Page 143
3.7.3 Entropy balance and pointwise convergence of the temperature ......Page 146
3.7.4 Pointwise convergence of the densities ......Page 150
3.7.5 Oscillations defect measure ......Page 155
3.8 Regularity properties of the weak solutions ......Page 159
4 Asymptotic Analysis – An Introduction......Page 164
4.1 Scaling and scaled equations ......Page 166
4.2 Low Mach number limits......Page 168
4.3 Strongly stratified flows ......Page 170
4.4.1 Low stratification ......Page 172
4.4.2 Strong stratification ......Page 174
4.4.3 Attenuation of acoustic waves ......Page 175
4.5 Acoustic analogies ......Page 176
4.6 Initial data ......Page 178
4.7 A general approach to singular limits for the full Navier-Stokes-Fourier system ......Page 179
5 Singular Limits – Low Stratification......Page 184
5.1.1 Hypotheses ......Page 187
5.1.2 Global-in-time solutions ......Page 189
5.2.1 Conservation of total mass ......Page 190
5.2.2 Total dissipation balance and related estimates ......Page 191
5.2.3 Uniform estimates......Page 194
5.3 Convergence ......Page 197
5.3.1 Equation of continuity ......Page 198
5.3.2 Entropy balance ......Page 199
5.3.3 Momentum equation......Page 203
5.4 Convergence of the convective term ......Page 206
5.4.1 Helmholtz decomposition ......Page 207
5.4.2 Compactness of the solenoidal part ......Page 208
5.4.3 Acoustic equation ......Page 209
5.4.4 Formal analysis of the acoustic equation ......Page 212
5.4.5 Spectral analysis of the wave operator ......Page 214
5.4.6 Reduction to a finite number of modes ......Page 215
5.4.7 Weak limit of the convective term – time lifting ......Page 217
5.5.1 Weak formulation of the target problem ......Page 220
5.5.2 Main result ......Page 222
5.5.4 Energy inequality for the limit system ......Page 223
6.1 Motivation ......Page 232
6.2.1 Field equations ......Page 233
6.2.2 Constitutive relations ......Page 234
6.2.3 Scaling ......Page 236
6.3.2 Solutions to the primitive system ......Page 238
6.3.3 Main result ......Page 240
6.4.1 Dissipation equation, energy estimates ......Page 243
6.4.2 Pressure estimates ......Page 248
6.5.2 Determining the pressure ......Page 251
6.5.3 Driving force ......Page 254
6.5.4 Momentum equation......Page 256
6.6.1 Acoustic equation ......Page 257
6.6.2 Spectral analysis of the wave operator ......Page 259
6.6.3 Convergence of the convective term ......Page 261
6.7 Asymptotic limit in entropy balance ......Page 265
7 Interaction of Acoustic Waves with Boundary......Page 268
7.1.1 Field equations ......Page 270
7.1.2 Physical domain and boundary conditions ......Page 272
7.2.1 Preliminaries – global existence ......Page 273
7.2.2 Compactness of the family of velocities ......Page 275
7.3 Uniform estimates ......Page 276
7.4.1 Acoustic equation ......Page 278
7.4.2 Spectral analysis of the acoustic operator ......Page 281
7.5.1 Compactness of the solenoidal component ......Page 290
7.5.2 Reduction to a finite number of modes ......Page 291
7.5.3 Strong convergence ......Page 292
8.1 Primitive system ......Page 298
8.2.1 Estimates based on the hypothesis of thermodynamic stability ......Page 301
8.2.2 Estimates based on the specific form of constitutive relations ......Page 303
8.3 Acoustic equation ......Page 305
8.4.1 Uniformestimates ......Page 308
8.4.2 Regularization ......Page 311
8.4.3 Extension to the whole space \mathbb{R}3......Page 313
8.5 Dispersive estimates and time decay of acoustic waves ......Page 314
8.6 Conclusion –main result ......Page 319
9 Acoustic Analogies......Page 322
9.1 Asymptotic analysis and the limit system ......Page 323
9.2 Acoustic equation revisited ......Page 324
9.3 Two-scale convergence ......Page 328
9.3.1 Approximate methods ......Page 332
9.4.1 Ill-prepared data ......Page 333
9.4.2 Well-prepared data ......Page 334
9.5 Concluding remarks ......Page 337
10.1 Mollifiers ......Page 340
10.2 Basic properties of some elliptic operators ......Page 341
10.2.1 A priori estimates ......Page 342
10.2.2 Fredholm alternative......Page 344
10.2.3 Spectrum of a generalized Laplacian ......Page 346
10.3 Normal traces ......Page 348
10.4 Singular and weakly singular operators ......Page 351
10.5 The inverse of the div-operator (Bogovskiifs formula) ......Page 352
10.6 Helmholtz decomposition ......Page 360
10.7 Function spaces of hydrodynamics ......Page 362
10.8 Poincaré type inequalities......Page 364
10.9 Korn type inequalities ......Page 366
10.10 Estimating ∇u by means of div_xu and curl_xu......Page 370
10.11 Weak convergence and monotone functions ......Page 371
10.12 Weak convergence and convex functions ......Page 375
10.13 Div-Curl lemma ......Page 378
10.14 Maximal regularity for parabolic equations ......Page 380
10.15 Quasilinear parabolic equations ......Page 382
10.16 Basic properties of the Riesz transform and related operators ......Page 384
10.17 Commutators involving Riesz operators ......Page 387
10.18 Renormalized solutions to the equation of continuity ......Page 389
11.1 Fluid flow modeling ......Page 396
11.2 Mathematical theory of weak solutions ......Page 397
11.4 Analysis of singular limits ......Page 398
11.5 Propagation of acoustic waves ......Page 399
Bibliography ......Page 400
Index ......Page 414
