Birkhäuser, Basel, Boston, Berlin, 2009. 419 p. ISBN:3764388420Many 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 fluidsContents Preface Acknowledgement Notation, Definitions, and Function Spaces Notation Differentialoperators Functionspaces Sobolevspaces Fouriertransform Weak convergence of integrable functions Non-negative Borel measures Parametrized (Young) measures Fluid Flow Modeling Fluidsincontinuummechanics Balancelaws Fieldequations Conservationofmass Balance of linear momentum Totalenergy Entropy Constitutiverelations Molecular energy and transport terms Stateequations Effect of thermal radiation Typical values of some physical coefficients Weak Solutions, A Priori Estimates Weakformulation Equationofcontinuity Balance of linear momentum Balance of total energy Entropyproduction Constitutive relations Aprioriestimates Totalmassconservation Energyestimates Estimates based on the Second law of thermodynamics Positivity of the absolute temperature Pressureestimates Pressure estimates, an alternative approach Existence Theory Hypotheses Structural properties of constitutive functions Mainexistenceresult Approximationscheme Solvability of the approximate system Approximate continuity equation Approximate internal energy equation Local solvability of the approximate problem Uniform estimates and global existence Faedo-Galerkinlimit Estimates independent of the dimension of Faedo-Galerkin approximations Limit passage in the approximate continuity equation Strong convergence of the approximate temperatures and the limit in the entropy equation Limit in the approximate momentum equation The limit system resulting from the Faedo-Galerkinapproximation The entropy production rate represented byapositivemeasure Artificialdiffusionlimit Uniform estimates and limit in the approximate continuity equation Entropy balance and strong convergence of the approximatetemperatures Uniformpressureestimates Limit in the approximate momentum equation andintheenergybalance Strong convergence of the densities Artificial diffusion asymptotic limit anishingartificialpressure Uniformestimates Asymptotic limit for vanishing artificial pressure Entropy balance and pointwise convergence ofthetemperature Pointwise convergence of the densities Oscillations defect measure Regularity properties of the weak solutions Asymptotic Analysis – An Introduction Scalingandscaledequations LowMachnumberlimits Strongly stratified flows Acousticwaves Lowstratification Strongstratification Attenuation of acoustic waves Acousticanalogies Initialdata A general approach to singular limits for thefullNavier-Stokes-Fouriersystem Singular Limits – Low Stratification Hypotheses and global existence for the primitive system Hypotheses Global-in-timesolutions Dissipation equation, uniform estimates Conservationoftotalmass Total dissipation balance and related estimates Uniformestimates Convergence Equationofcontinuity Entropybalance Momentumequation Convergence of the convective term Helmholtzdecomposition Compactness of the solenoidal part Acousticequation Formal analysis of the acoustic equation Spectral analysis of the wave operator Reduction to a finite number of modes Weak limit of the convective term – time lifting Conclusion–mainresult Weak formulation of the target problem Mainresult Determining the initial temperature distribution Energy inequality for the limit system Stratified Fluids Motivation Primitivesystem Fieldequations Constitutive relations Scaling Asymptoticlimit Staticstates Solutionstotheprimitivesystem Mainresult Uniformestimates Dissipation equation, energy estimates Pressureestimates Convergencetowardsthetargetsystem Anelasticconstraint Determiningthepressure Drivingforce Momentumequation Analysisofacousticwaves Acousticequation Spectral analysis of the wave operator Convergence of the convective term Asymptotic limit in entropy balance Interaction of Acoustic Waves with Boundary Problemformulation Fieldequations Physical domain and boundary conditions Mainresult Preliminaries – global existence Compactness of the family of velocities Uniformestimates Analysisofacousticwaves Acousticequation Spectral analysis of the acoustic operator Strong convergence of the velocity field Compactness of the solenoidal component Reduction to a finite number of modes Strongconvergence ProblemsonLargeDomains Primitivesystem Uniformestimates Estimates based on the hypothesis of thermodynamic stability Estimates based on the specific form of constitutiverelations Acousticequation Regularization and extension to R Uniformestimates Regularization Extension to the whole space R Dispersive estimates and time decay of acoustic waves Conclusion–mainresult Acoustic Analogies Asymptotic analysis and the limit system Acousticequationrevisited Two-scaleconvergence Approximate methods Lighthill’s acoustic analogy in the low Machnumberregime ll-prepareddata Well-prepared data Concludingremarks Appendix Mollifiers Basic properties of some elliptic operators Aprioriestimates Fredholmalternative Spectrum of a generalized Laplacian Normaltraces Singular and weakly singular operators The inverse of the div-operator (Bogovskii’s formula) Helmholtz decomposition Function spaces of hydrodynamics Poincar´etypeinequalities Korntypeinequalities Estimating ∇u by means of div x u and curl x u Weak convergence and monotone functions Weak convergence and convex functions Div-Curllemma Maximal regularity for parabolic equations Quasilinear parabolic equations Basic properties of the Riesz transform and relatedoperators Commutators involving Riesz operators Renormalized solutions to the equation of continuity Bibliographical Remarks Fluidflowmodeling Mathematical theory of weak solutions Existencetheory Analysisofsingularlimits Propagation of acoustic waves Bibliography Index
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