EXPONENTIAL DICHOTOMIES OF STOCHASTIC SOBOLEV TYPE EQUATIONS

Abstract

The article is of a review nature and contains results on the study of the stability of Sobolev type stochastic linear equations in terms of stable and unstable invariant spaces and exponential dichotomies. The article considers stochastic analogues of the Barenblatt -- Zheltov -- Kochina equation for the pressure of a fluid filtering in a fractured porous medium, the Oskolkov linear equation of plane-parallel flows of a viscoelastic fluid, the Dzektser equation describing the evolution of the free surface of a filtering liquid, the Ginzburg -- Landau equation, which models the conductivity in a magnetic field. These equations can be considered as special cases of the stochastic Sobolev-type equations, where the stochastic K-process acts as the required quantity, and its derivative is understood as the Nelson -- Glicklich derivative. The paper presents results on the existence of stable and unstable invariant spaces of stochastic equations that are Barenblatt -- Zheltov -- Kochina, Oskolkov, Dzektser and Ginzburg -- Landau equations. The general scheme of a numerical algorithm for finding stable and unstable solutions to these equations is described, and the results of computational experiments are presented.