Singular Limits in Thermodynamics of Viscous Fluids

Preface.- 1 Fluid flow modeling.- 1.1 Field equations of continuum fluid mechanics.- 1.2 Constitutive relations.- 2 Mathematical theory of weak solutions.- 2.1 Variational formulation.- 2.2 A priori estimates.- 3 Existence theory.- 3.1 Hypotheses.- 3.2 Structural properties of constitutive functions.- 3.3 Main existence result.- 3.4 Solvability of the approximate system.- 3.5 Limit in the Faedo-Galerkin approximation scheme.- 3.6 Artificial diffusion limit.- 3.7 Vanishing artificial pressure.- 3.8 Regularity properties of weak solutions.- 4 Asymptotic analysis - an introduction.- 4.1 Scaling and scaled equations.- 4.2 Low Mach number limit.- 4.3 Strongly satisfied flows.- 4.4 Acoustic waves.- 5 Singular limits - low stratification.- 5.1 Hypotheses and global existence for the primitive system.- 5.2 Dissipation equation, uniform solutions.- 5.3 Convergence.- 5.4 Acoustiv waves.- 5.5 Conclusion - main result.- 6 Stratified fluids.- 6.1 Motivation.- 6.2 Primitive system.- 6.3 Asymptotic limit.- 6.4 Uniform estimates.- 6.5 Convergence towards the target system.- 6.6 Analysis of the acoustic waves.- 6.7 Asymptotic limit in the entropy balance.- 7 Refined analysis of the acoustic waves.- 7.1 Problem formulation.- 7.2 Main result.- 7.3 Uniform estimates.- 7.4 Analysis of the acoustic waves.- 7.5 Strong convergence of the velocity field.- 8 Appendix.- 8.1 Quasilinear parabolic equations.- 8.2 Mollifiers.- 8.3 The normal traces.- 8.4 The Bogovskii Operator.- 8.5 Maximal regularity to parabolic equations.- 8.6 Korn and Poincaré type inequalities.- 8.7 Radon measures.- 8.8 Weak convergence, monotone and convex functions.- 8.9 Fourier and the Riesz transforms.- 8.10 Div-Curl lemma and commutators involving the Riesz operators.- 8.11 Renormalized solutions to the continuity equation.- 9 Bibliographic remarks 9.1 Fluid flow modeling.- 9.2 Mathematical theory of the weak solutions.- 9.3 Singular limits.