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Singular Limits in Thermodynamics of Viscous Fluids
Eduard Feireisl, Antonín Novotný
Verlag Birkhäuser Basel, 2009
ISBN 9783764388430 , 382 Seiten
Format PDF, OL
Kopierschutz Wasserzeichen
Geräte
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
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