Asymptotically Autonomous Robustness of Non-autonomous Random Attractors for Stochastic Convective Brinkman-Forchheimer Equations on ℝ3

Abstract

Abstract This article is concerned with the asymptotically autonomous robustness (almost surely and in probability) of random attractors for stochastic version of 3D convective Brinkman-Forchheimer (CBF) equations defined on $\mathbb {R}^{3}$: $$ \begin{align*} &\frac{\partial\boldsymbol{v}}{\partial\mathrm{t}}-\mu\Delta\boldsymbol{v}+(\boldsymbol{v}\cdot\nabla)\boldsymbol{v}+\alpha\boldsymbol{v}+\beta|\boldsymbol{v}|^{r-1}\boldsymbol{v}+\nabla{p}=\boldsymbol{f}+``\mbox{stochastic terms}",\quad\nabla\cdot\boldsymbol{v}=0,\end{align*}$$where $\mu ,\alpha ,\beta > 0$, $r\geq 1$ and $\boldsymbol {f}(\cdot )$ is a given time-dependent external force field. Our goal is to study the asymptotically autonomous robustness for 3D stochastic CBF equations perturbed by a linear multiplicative or additive noise when time-dependent forcing converges towards a time-independent function. The main procedure to achieve our goal is how to justify that the usual pullback asymptotic compactness of the solution operators is uniform on some uniformly tempered universes over an infinite time-interval $(-\infty ,\tau ]$. This can be done by showing the backward uniform “tail-smallness” and “flattening-property” of the solutions over $(-\infty ,\tau ]$.