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Review Articles

On a class of Sobolev-type equations

T. G. Sukacheva, A. O. Kondyukov

Novgorod State University, Velikiy Novgorod, Russian Federation

Abstract: The article surveys the works of T. G. Sukacheva and her students studying the models of incompressible viscoelastic Kelvin–Voigt fluids in the framework of the theory of semilinear Sobolev-type equations. We focus on the unstable case because of greater generality. The idea is illustrated by an example: the non-stationary thermoconvection problem for the order 0 Oskolkov model. Firstly, we study the abstract Cauchy problem for a semilinear nonautonomous Sobolev-type equation. Then, we treat the corresponding initial-boundary value problem as its concrete realization. We prove the existence and uniqueness of a solution to the stated problem. The solution itself is a quasi-stationary semi-trajectory. We describe the extended phase space of the problem. Other problems of hydrodynamics can also be investigated in this way: for instance, the linearized Oskolkov model, Taylor's problem, as well as some models describing the motion of an incompressible viscoelastic Kelvin–Voigt fluid in the magnetic field of the Earth.

Keywords: Sobolev type equations; incompressible viscoelastic fluids; relatively $p$-sectorial operators; extended phase spaces.

UDC: 517.9

MSC: 35K70

Received: 15.09.2014

Language: English

DOI: 10.14529/mmp140401