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Singular Limits in Thermodynamics of Viscous Fluids

:	Eduard Feireisl, Antonín Novotný
:	Singular Limits in Thermodynamics of Viscous Fluids
:	Birkhäuser Basel
:	9783764388430
:	1
:	CHF 124.10
:

:	Mechanik, Akustik
:	English

:	382
:	DRM
:	PC/MAC/eReader/Tablet
:	PDF

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.

	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