Large time behavior of the deterministic and stochastic 3D convective Brinkman-Forchheimer equations in periodic domains

TL;DR

This paper investigates the long-term dynamics of 3D convective Brinkman-Forchheimer equations, establishing the existence of attractors in both deterministic and stochastic settings, and analyzing their stability under small random perturbations.

Contribution

It proves the existence of global and random attractors for the 3D CBF equations and demonstrates the upper semicontinuity of these attractors as stochastic noise diminishes.

Findings

Existence of global attractors for deterministic 3D CBF equations.

Existence of random attractors for stochastic 3D CBF equations.

Upper semicontinuity of random attractors as noise approaches zero.

Abstract

The large time behavior of the deterministic and stochastic three dimensional convective Brinkman-Forchheimer (CBF) equations for $r\geq3$ ($r>3$, for any $\mu$ and $\beta$, and $r=3$ for $2\beta\mu\geq1$), in periodic domains is carried out in this work. Our first goal is to prove the existence of global attractors for the 3D deterministic CBF equations. Then, we show the existence of random attractors for the 3D stochastic CBF equations perturbed by small additive smooth noise. Finally, we establish the upper semicontinuity of random attractor for the 3D stochastic CBF equations (stability of attractors), when the coefficient of random perturbation approaches to zero.