Oskolkov Models and Sobolev-Type Equations

This article is a review of the works carried out by the author together with her students and devoted to the study of various Oskolkov models. Their distinctive feature is the use of the semigroup approach, which is the basis of the phase space method used widely in the theory of Sobolev-type equations. Various models of an incompressible viscoelastic fluid described by the Oskolkov equations are presented. The degenerate problem of magnetohydrodynamics, the problem of thermal convection, and the Taylor problem are considered as examples. The solvability of the corresponding initial-boundary value problems is investigated within the framework of the theory of Sobolev-type equations based on the theory for p -sectorial operators and degenerate semigroups of operators. An existence theorem is proved for a unique solution, which is a quasi-stationary semitrajectory, and a description of the extended phase space is obtained. The foundations of the theory of solvability of Sobolev-type equations were laid by Professor G.A. Sviridyuk. Then this theory, together with various applications, was successfully developed by his followers.