Long term behavior of 2D and 3D non-autonomous random convective Brinkman-Forchheimer equations driven by colored noise

Abstract

The long time behavior of Wong-Zakai approximations of 2D as well as 3D non-autonomous stochastic convective Brinkman-Forchheimer (CBF) equations with non-linear diffusion terms on bounded and unbounded (${\mathbb{R}}^d$ for $d=2,3$) domains is discussed in this work. To establish the existence of random pullback attractors, the concept of asymptotic compactness (AC) is used. In bounded domains, AC is proved via compact Sobolev embeddings. In unbounded domains, due to the lack of compact embeddings, the ideas of energy equations and uniform tail estimates are exploited to prove AC. In the literature, CBF equations are also known as \emph{Navier-Stokes equations (NSE) with damping}, and it is interesting to see that the modification in NSE by linear and nonlinear damping provides better results than that available for NSE. The presence of linear damping term helps to establish the results in the whole domain ${\mathbb{R}}^d$. The nonlinear damping term supports to obtain better results in 3D and also for a large class of nonlinear diffusion terms. Moreover, we prove the existence of a unique random pullback attractor for stochastic CBF equations with additive white noise. Finally, for additive as well as multiplicative noise case, we establish the convergence of solutions and upper semicontinuity of random pullback attractors for Wong-Zakai approximations of stochastic CBF equations towards the random pullback attractors for stochastic CBF equations when correlation time of colored noise converges to zero.

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