Asymptotically autonomous robustness in Probability of non-autonomous random attractors for stochastic convective Brinkman-Forchheimer equations on ${\mathbb{R}}^3$

This article is concerned with the \emph{asymptotically autonomous robustness} (almost surely and in probability) of non-autonomous random attractors for two stochastic versions of 3D convective Brinkman-Forchheimer (CBF) equations defined on the whole space ${\mathbb{R}}^3$: $$\frac{\mathrm{\partial }\mathit{v}}{\mathrm{\partial }t}-\mu \mathrm{\Delta }\mathit{v}+(\mathit{v}\cdot \mathrm{\nabla })\mathit{v}+\alpha \mathit{v}+\beta \mathrm{\mid }\mathit{v}{\mathrm{\mid }}^{r-1}\mathit{v}+\mathrm{\nabla }p=\mathit{f}(t)+\mathrm{‘}\mathrm{‘}stochasticterms\mathrm{"},\mathrm{\nabla }\cdot \mathit{v}=0,$$ with initial and boundary vanishing conditions, where $\mu ,\alpha ,\beta >0$, $r\ge 1$ and $\mathit{f}(\cdot )$ is a given time-dependent external force field. By the asymptotically autonomous robustness of a non-autonomous random attractor $\mathcal{A}=\{\mathcal{A}(\tau ,\omega ):\tau \in \mathbb{R},\omega \in \mathrm{\Omega }\}$ we mean its time-section $\mathcal{A}(\tau ,\omega )$ is robust to a time-independent random set as time $\tau$ tends to negative infinity according to the Hausdorff semi-distance of the underlying space. Our goal is to study this topic, almost surely and in probability, for the non-autonomous 3D CBF equations when the stochastic term is a linear multiplicative or additive noise, and the time-dependent forcing converges towards a time-independent function. Our main results contain two cases: i) $r\in (3,\mathrm{\infty })$ with any $\beta ,\mu >0$; ii) $r=3$ with $2\beta \mu \ge 1$. 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 \emph{uniformly} tempered universes over an \emph{infinite} time-interval $(-\mathrm{\infty },\tau ]$. This can be done by a method based on Kuratowski’s measure of noncompactness.