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Mathematical topics in fluid mechanics. Vol. 1: Incompressible models. (English) Zbl 0866.76002
Oxford Lecture Series in Mathematics and its Applications. 3. Oxford: Clarendon Press. xiv, 237 p. (1996).
The author presents various mathematical results on incompressible fluid models including density-dependent Navier-Stokes and Euler equations. A lot of results are new, and for each of these recent results the complete and self-contained proofs are given. The author does not demand from the reader technical prerequisites other than a basic training in nonlinear partial differential equations. Firstly, he recalls the fundamental equations modeling Newtonian fluids together with the basic approximated and simplified models. Then the most general existence results for density-dependent Navier-Stokes equations are presented along with the discussion on regularity and uniqueness, on stationary problems, and mentioning a number of open questions.
Deep analysis is applied to the classical Navier-Stokes equations for homogeneous fluids. Namely, the celebrated results of J. Leray concerning the global existence of weak solutions are deduced, and some recent regularity results in three dimensions are thoroughly treated as well. A great attention is devoted to the classical Euler equations. The author first recalls the state of the art of this model, then compares the multiple notions of weak solutions and shows their existence and uniqueness for the two-dimensional case. Next, he discusses the open question concerning a priori estimates in three dimensions and gives some examples. Finally, some approaches to proving global existence in a supergeneralized weak sense are studied, and density-dependent Euler equations and hydrostatic approximation models are analyzed. In the appendices various useful technical results are presented for the reader’s convenience.
Summarizing the impression of this interesting book, it is worth pointing out that it is written in an easy-reading fashion along with the deep and comprehensive analysis of the topics which are at the highest level of the importance in the modern theory of nonlinear differential equations.

MSC:
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
76B47 Vortex flows for incompressible inviscid fluids
35Q30 Navier-Stokes equations
35Q35 PDEs in connection with fluid mechanics
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