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Operator methods in hydrodynamics. Evolutionary and spectral problems. (Операторные методы в линейной гидродинамике. Ѐволюционные и спектральные задачи.) (Russian. English summary) Zbl 0681.76001

Moskva: Nauka. 416 p. R. 6.10 (1989).
Summary: Problems on the motion of rigid bodies with cavities filled with liquid are of the classical type. The beginning of their investigation goes back to J. Stokes, H. Helmholtz, G. Lamb, N. E. Zhukovsky. Recently these problems have become actual in connection with the needs of aviation, tanker, rocket and cosmic technique. The book contains a broad exposition of the results, which have been obtained up till now by means of operator methods and problems of the dynamics of bodies having cavities filled with liquid. In the course of investigating these problems, different linear operators appear. The study of the interaction of these operators and their corresponding evolution and spectral problems is the main purpose of this book.
The readers are not supposed to be familiar with the methods of functional analysis that is why in the first part of the book the main facts of the linear operators theory used in the course of studying linearized problems of hydrodynamics are presented without proofs. These facts include elements of the generalized function theory, the theory of self-adjoint operators in a Hilbert space and in a space with an indefinite metric, the theory of evolution equations and asymptotic methods of their solutions, the spectral theory of operator bundles and so on. Thus, the first part of the book may serve as a reference book on a wide range of questions of functional analysis. Naturally, domains with non-smooth boundaries appear in the problems of hydrodynamics (containers with edges, partly filled with liquid cavities and so on).
An abstract scheme allowing to establish the existence of generalized and weak solutions of boundary value problems in domains with non-smooth boundaries under discontinuous boundary conditions is given in § 1.3.
The authors follow the ideology that metrics in different spaces of fields of the fluids motion speeds must have a physical sense. For an ideal liquid such metric is generated by a quadratic form of kinetic energy and for a viscous liquid this metric is generated by the same form and the form of energy dissipation speed. All the other spaces of fields are subspaces of Hilbert spaces generated by the just mentioned forms. Chapter 2 is devoted to the description of widely used domains of definition of the vector analysis main operators in the frame of the above mentioned spaces of fields.
The system of hydrodynamics equation is not of Cauchy-Kovalevskaya type as the time derivative of pressure in not included in them. In classical hydrodynamics to exclude the pressure the operator rot is applied in both sides of the equation, however, this doesn’t improve the quality of the system. Independently E. Hopf and S. G. Krein, and also S. L. Sobolev used the orthogonal projection method for the solenoidal fields subspace satisfying the required boundary conditions. This projection method is being used in all parts of the book.
The second part of the book is devoted to the study of the motion of bodies with cavities containing ideal liquid, part III deals with cavities containing viscous fluid. The authors consider the motion of fluids entirely or partly filling a rigid body cavity, the body being immoved or slowly oscillating around a fixed point, the motions may be similar to rotations around a fixed axis. The main results are concerned with the proofs of the existence of solutions of the corresponding initial boundary-value problems, with the study of normal oscillation properties (of inner and surface waves), with the structure of the spectrum of their frequencies, and obtaining of asymptotic formula.
In the main part of the book the body is considered to be absolutely rigid. However in chapter 4 the motion of an ideal fluid in an immovable container with elastic bottoms has been investigated.
Apart from the main problems in chapters 6 and 7 the authors consider the convective motion of a viscous fluid entirely or partly filling a rigid body’s cavity. And finally in chapters 4 and 8 the authors investigate the capillary viscous fluid oscillation.
The authors are famous specialists of the problems under discussion in this book. The authors works carried out in the last 25-35 years constitute the basis of the exposition of the hydrodynamic problems. In its contents the book does not cross with other books on hydrodynamics. The book is particularly useful for mechanicians interested in the theory of motions of bodies with fluid fillings and for mathematicians who wish to master the operator methods in this theory.

MSC:

76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
47N50 Applications of operator theory in the physical sciences
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