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 of how to get from the compressible Navier-Stokes-Fourier system several less complex systems of partial differential equations, used, for example, in meteorology or astrophysics. However, the book also contains a detailed introduction to the modelling in mechanics and thermodynamics of fluids from the viewpoint of continuum mechanics. It also includes the proof of the existence of a weak solution to the full Navier-Stokes-Fourier system, which has not been presented in such a detailed form before. All limits are considered for these solutions whose existence is ensured for large data (i.e. without any smallness assumptions on the data or length of the time interval). Rewriting the full system to the dimensionless form, limits when the Mach, Froude or Péclet numbers tend to zero are studied. According to the rate of convergence, either the Oberbeck-Boussinesq or unelasting approximations are obtained, in bounded domains for thermally insulated domains with either slip or no-slip boundary conditions for velocity. In the case of unbounded domain, these results are obtained independently of boundary conditions. The main difficulty is connected with acoustic waves, which appear due to the compressibility of the fluid. The book is intended for specialists in the field; however, it can also be used for doctoral students and young researchers who want to start to work with the mathematical theory of compressible fluids and their asymptotic limits.

Reviewer:

mpok