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Mathematical theory of compressible viscous fluids. Analysis and numerics. (English) Zbl 1356.76001

Advances in Mathematical Fluid Mechanics. Lecture Notes in Mathematical Fluid Mechanics. Basel: Birkhäuser/Springer (ISBN 978-3-319-44834-3/pbk; 978-3-319-44835-0/ebook). xii, 186 p. (2016).
The authors present the mathematical theory of compressible barotropic viscous fluids in the framework of weak solutions introduced by P.-L.Lions [Mathematical topics in fluid mechanics. Vol. 2: Compressible models. Oxford: Clarendon Press (1998; Zbl 0908.76004)] and extended by the first author [Dynamics of viscous compressible fluids. Oxford: Oxford University Press (2004; Zbl 1080.76001)].
After a short introduction presenting the most important tools (including familiar and also more advanced items), the first part of the monograph is devoted to the existence of a weak solution for a compressible barotropic Navier-Stokes system. In this part, the “simple” case of pressure \(p(\varrho)\) where \(s\to p(s)\) is smooth, positive and strictly monotone is considered. Namely, one assumes that \[ \lim_{\varrho\to\infty}\frac{p'(\varrho)}{\varrho^{\gamma-1}}=p_\infty>0, \] for some \(\gamma>3/2\). Then all the necessary a priori estimates can be proved for density, energy and pressure, leading to a complete proof of weak sequential stability.
In the second part, an alternative (and constructive) proof of the existence of weak solutions is presented, relying on a numerical approximation scheme: a time discretization is coupled to a mixed finite element, finite volume discretization to solve the resulting stationary problems. Just mention that, in order to implement the method, a strengthened condition \(\gamma>3\) is necessary (at least if one desires to keep the constructive character of the proof). Then, stability, consistency and finally convergence of the scheme are explained in detail.
In Part 3, the previous theory is extended to more general pressure laws, including no-monotone functions \(p(\varrho)\). In this case, more advanced notions as oscillation defect measures have to be introduced.
Finally, the last section is devoted to a brief point on present research and suggestions for related reading.
This very attractive, short and self-contained monograph is perfectly suitable for a beginning graduate student who wants to learn about mathematical fluid mechanics, as a preparation for more advanced textbooks covering polytropic situations [the first author, loc. cit.] and singular limits [the first author and A. Novotný, Singular limits in thermodynamics of viscous fluids. Basel: Birkhäuser (2009; Zbl 1176.35126)].

MSC:

76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
76N15 Gas dynamics (general theory)
35Q30 Navier-Stokes equations
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