Large Time Behavior of Deterministic and Stochastic 3D Convective Brinkman-Forchheimer Equations in Periodic Domains

Abstract

The large time behavior of deterministic and stochastic three dimensional convective Brinkman-Forchheimer (CBF) equations $$\begin{aligned} \partial _t{\varvec{u}}-\mu \Delta {\varvec{u}}+({\varvec{u}}\cdot \nabla ){\varvec{u}}+\alpha {\varvec{u}}+\beta |{\varvec{u}}|^{r-1}{\varvec{u}}+\nabla p={\varvec{f}},\ \nabla \cdot {\varvec{u}}=0, \end{aligned}$$ ∂ t u - μ Δ u + ( u · ∇ ) u + α u + β | u | r - 1 u + ∇ p = f , ∇ · u = 0 , for $$r\ge 3$$ r ≥ 3 ( $$\mu ,\beta >0$$ μ , β > 0 for $$r>3$$ r > 3 and $$2\beta \mu \ge 1$$ 2 β μ ≥ 1 for $$r=3$$ r = 3 ), 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. Furthermore, we establish the upper semicontinuity of random attractors for the 3D stochastic CBF equations (stability of attractors), when the coefficient of random perturbation approaches to zero. Finally, we address the existence and uniqueness of invariant measures of 3D stochastic CBF equations.