Existence and upper semicontinuity of random attractors for the 2D stochastic convective Brinkman-Forchheimer equations in bounded domains

TL;DR

This paper proves the existence and upper semicontinuity of random attractors for the 2D stochastic convective Brinkman-Forchheimer equations in bounded domains, demonstrating long-term behavior and invariant measures under small noise perturbations.

Contribution

It establishes the existence of random attractors in both Hilbert and more regular spaces for the 2D SCBF equations, including upper semicontinuity as noise diminishes.

Findings

Existence of random attractors in  for 2D SCBF equations.

Upper semicontinuity of attractors as noise coefficient approaches zero.

Existence of invariant measures for the stochastic flow.

Abstract

In this work, we discuss the large time behavior of the solutions of the two dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations in bounded domains. Under the functional setting $\V\hookrightarrow\H\hookrightarrow\V'$, where $\H$ and $\V$ are appropriate separable Hilbert spaces and the embedding $\V\hookrightarrow\H$ is compact, we establish the existence of random attractors in $\H$ for the stochastic flow generated by the 2D SCBF equations perturbed by small additive noise. We prove the upper semicontinuity of the random attractors for the 2D SCBF equations in $\H$, when the coefficient of random term approaches zero. Moreover, we obtain the existence of random attractors in a more regular space $\V$, using the pullback flattening property. The existence of random attractors ensures the existence of invariant compact random set and hence we show the existence of an invariant measure for the 2D SCBF equations.