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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.
List of contents
1 Fluid flow modeling.- 2 Mathematical theory of weak solutions.- 3 Existence theory.- 4 Asymptotic analysis - an introduction.- 5 Singular limits - low stratification.- 6 Stratified fluids.- 7 Refined analysis of the acoustic waves.- 8 Appendix.- 9 Bibliographic remarks.
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From the reviews:
"This book is a very interesting contribution to the mathematical theory of partial differential equations describing the flow of compressible heat conducting fluids together with their singular limits. ... The book is very interesting and important. It can be recommended not only to specialists in the field, but it can also be used for doctoral students and young researches who want to start to work in the mathematical theory of compressible fluids and their asymptotic limits." (Milan Pokorny, Zentralblatt MATH, Vol. 1176, 2010)
"The authors start from the observation that ... mathematical models used in fluid mechanics rely on formal asymptotic analysis of more complex systems. The purpose of their book is to give rigorous proofs of several of these asymptotics. ... This book is of the highest quality from every point of view. ... It is extremely well organized, and very well written. It is a landmark for researchers in mathematical fluid dynamics, especially those interested in the physical meaning of the equations and statements." (Denis Serre, Mathematical Reviews, Issue 2011 b)
