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Boundary value problems in the mechanics of inhomogeneous fluids. (Краевые задачи механики неоднородных жидкостей.) (Russian) Zbl 0568.76001

Akademiya Nauk SSSR, Sibirskoe Otdelenie Institut Gidrodinamiki im. M. A. Lavrent’eva. Novosibirsk; Izdatel’stvo “Nauka” Sibirskoe Otdelenie. 320 p. R. 3.40 (1983).
The paper is a monograph devoted to the boundary value problems appearing in the hydrodynamics of homogeneous and nonhomogeneous fluids. A great part of the exposed results is the fruit of the proper scientific investigations of the authors, obtained especially in the Hydrodynamical Institute of the Soviet Scientific Academy of Novosibirsk.
In the introductory chapter the general principles are exposed, by means of which the general equations of motion of a continuous medium can be deduced; the equations of motion of inviscid or viscous, compressible or incompressible, homogeneous or nonhomogeneous fluids are deduced as particular cases from the general ones. There are also presented the principal results and methods of functional analysis used in the study of the equations of motion. The proper material of the book is presented in the following four chapters: the one-dimensional nonstationary motion of a viscous gas, the Navier-Stokes equations of an incompressible, nonhomogeneous, viscous fluid, Euler’s equations of an incompressible fluid and the two phase flow of two immiscible fluids in a porous medium.
The point of view is purely theoretical; the following questions are considered in each case, with some differences from one problem to another: the initial boundary value problem, the a priori evaluations of the solution and of their derivatives, the existence and uniqueness conditions of the generalised or classical solutions, and effective methods how to find them. Each problem is put in its historical context with critical considerations regarding the applied methods and the obtained results up to the recent ones, where the authors have their own contributions. Each chapter ends with a short enumeration of the difficult problems unsolved till now and, sometimes, of suggestions and opinions about the possible solutions.
In our opinion the paper is useful both for beginners and for specialists; they will find a synthesis of the most recent results in the studied domain, systematic and critical methods used now in hydrodynamics and a clear exposition of the possible new investigations.

MSC:

76-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to fluid mechanics
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
35Q30 Navier-Stokes equations
35Q31 Euler equations
76D05 Navier-Stokes equations for incompressible viscous fluids
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics