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Feireisl E., Novotný A. Singular Limits in Thermodynamics of Viscous Fluids

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Feireisl E., Novotný A. Singular Limits in Thermodynamics of Viscous Fluids
Birkhäuser, Basel, Boston, Berlin, 2009. 419 p. ISBN:3764388420
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
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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