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Boundary value problems in mechanics of nonhomogeneous fluids. Transl. from the Russian. (English) Zbl 0696.76001
Studies in Mathematics and Its Applications, 22. Amsterdam etc.: North- Holland. xii, 309 p. $ 85.25; Dfl. 175.00 (1990).
[The Russian original (1983) has been reviewed in Zbl 0568.76001.]
This book studies the correctness of the initial boundary value problems for systems of partial differential equations describing the flows of viscous gas, density-inhomogeneous fluid and the filtration of a multiphase mixture in a porous medium.
The material of the book is presented in five chapters. In the first part of Chapter 1 the authors present some models used in hydrodynamics: models of viscous perfect polytropic gas, heterogeneous viscous incompressible fluid, ideal, incompressible fluid, models of multicomponent and multiphase mixtures, model of a medium with intrinsic degrees of freedom, equations of filtration of two immiscible incompressible fluids. The second part of Chapter 1 contains some results of functional analysis and theory of differential equations.
The second Chapter presents the problems of the existence of the solutions of the boundary problems for a system of the Navier-Stokes equations. Besides proving a unique solvability the solution behaviour at an infinitely growing time is studied.
The third chapter considers the model of a nonhomogeneous viscous, incompressible liquid. Interest in the model of nonhomogeneous liquid is conditioned by its importance for applied fields of hydrodynamics, such as oceanology and hydrology.
Chapter 4 contains a classical hydrodynamical model: the Euler equations for an ideal liquid. The main result of Chapter 4 consists in proving the solvability of the problem of an ideal liquid flowing through a bounded domain, when on the portion of in-flow the velocity is given, while on the portion of out-flow only its normal component is given.
The last Chapter is devoted to the investigation of filtration equations for two immiscible incompressible liquids in a porous medium. This model is reduced to a system of two second-order equations, one being elliptic and the second being parabolic. The results presented in Chapter 5 include the proof of the existence of generalized solutions, the study of their differential properties and their uniqueness.
Chapters 2-5 end with some unsolvable problems which in the authors’ opinion are of scientific interest.
Reviewer: R.Stavre

76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
35Q30 Navier-Stokes equations
35Q99 Partial differential equations of mathematical physics and other areas of application