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Singular limits in thermodynamics of viscous fluids. (English) Zbl 1176.35126

Advances in Mathematical Fluid Mechanics. Basel: Birkhäuser (ISBN 978-3-7643-8842-3/hbk; 978-3-7643-8843-0/ebook). xxxvi, 382 p. (2009).
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 main aim is to provide mathematically rigorous arguments how to get from the compressible Navier-Stokes-Fourier system several less complex systems of partial differential equations used e.g. in meteorology or astrophysics. However, the book contains also a detailed introduction to the modelling in mechanics and thermodynamics of fluids from the viewpoint of continuum physics.
The book consists of 12 parts. In the first part (Chapter 0), basic notation and general mathematical tools are introduced. Chapter 1 is devoted to the description of the heat conducting newtonian compressible fluid. Next chapter contains weak formulation of the balance laws, together with formal a priori estimates. Chapter 3 contains the existence proof of a weak solution to the above mentioned system of partial differential equations. This result has been used many times before, however, a detailed rigorous proof of it was missing in the literature, due to its length and technical complications. Thus this book fills this gap.
Next chapter contains a brief introduction to singular limits. The Navier-Stokes-Fourier system is rewritten into dimensionless form using several characteristic numbers (Strouhal, Mach, Reynolds, Froude, Péclet numbers and heat release parameter). Passing with Mach (and possibly other) number to zero we get formally several easier systems of partial differential equations. The rest of the book is devoted to rigorous verification of these limits. The main difficulty in the analysis of the limit process is connected with the acoustic waves which are closely related to compressibility of the fluid.
Chapter 5 contains study of the limit in the case of low stratification, i.e. the Mach number \(Ma = \varepsilon\), the Froude number \(\text{Fr}= \sqrt{\varepsilon}\) for \(\varepsilon \to 0^+\), together with the heat insulation boundary condition for the temperature (zero heat flux) and slip boundary condition for the velocity. It is proved that the solution converges to the solution of the so-called Oberbeck-Boussinesq approximation. The limit is considered for solutions whose existence is ensured for large data, i.e. without any smallness assumptions on the data or length of the time interval. The data are ill-prepared which leads to further complications. The same holds also for other limits.
Next chapter is devoted to the limit for stratified fluids, i.e. \(\text{Ma}= \text{Fr}=\varepsilon\) and the Péclet number \(\text{Pe}= \varepsilon^2\) for \(\varepsilon \to 0^+\). The domain is a two-dimension flat torus with slip boundary condition on the top and bottom and heat insulation condition on the bottom and radiative condition on the top, together with periodic conditions on the sides. It is shown that the solution converges to a solution of the so-called anelastic approximation. Next chapter contains analysis of the acoustic equations for low stratificatied fluids under no-slip boundary condition for the velocity. It is shown that the velocity converges strongly in \(L^2\). The other parts of the proof from Chapter 5 would be the same.
Chapter 8 deals with similar problem in unbounded domains, where even the domains may change with \(\varepsilon\). Local strong convergence of the velocity is proved provided the domains satisfy a condition (Property \(L\)) which ensures that the acoustic wave initiated in a bounded subdomain cannot reach the boundary. The result is independent of the boundary conditions imposed on the velocity. Next chapter deals with two-scale convergence of the acoustic equation. Last two chapters contain advanced mathematical tools used throughout the book and bibliographical remarks.
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.

MSC:

35Q30 Navier-Stokes equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
80A22 Stefan problems, phase changes, etc.
35B45 A priori estimates in context of PDEs
76Q05 Hydro- and aero-acoustics
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