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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.

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: Ruxandra Stavre (Bucureşti)

### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |

35Q30 | Navier-Stokes equations |

35Q31 | Euler equations |

86A05 | Hydrology, hydrography, oceanography |