RANDOM DYNAMICS OF CONVECTIVE BRINKMANFORCHHEIMER EQUATIONS

Abstract

For many years, dynamic systems theory has been a very active topic of study in Mathematics and related sciences. One of the major problems in the study of evolution equations of Mathematical Physics is the investigation of the large time behavior of the solutions of these equations. Note that it is not always possible to find an explicit solution of a simple ordinary differential equation (ODE) or a partial differential equation (PDE) also. Therefore one needs a qualitative theory to capture the long time behavior of the solution of a ODE or PDE. For a number of evolution equations of Mathematical Physics, it was shown that the long time behavior of their solutions is characterized by attractors (cf. [44]). The theories and applications of attractors for deterministic dynamical systems can be referred to some prominent works [8, 38, 44, 80, 119, 133] and many others. Since evolution equations arriving from Physics and other fields of science are often not deterministic, the concept of large time behavior of stochastic equations driven by uncertain forcing, known as random (pullback) attractors, has been studied in the works [23, 50, 51, 141], etc. We remark that in order to study pathwise attractors, one needs to convert a stochastic system into a pathwise deterministic one, which is possible for additive and linear multiplicative white noises. Therefore, in this thesis, we mostly consider either additive white noise or linear multiplicative white noise.