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

Abstract

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 ${\mathbb{R}}^3$: $$\frac{\partial\boldsymbol{v}}{\partial t}-μΔ\boldsymbol{v}+(\boldsymbol{v}\cdot\nabla)\boldsymbol{v} +α\boldsymbol{v}+ β|\boldsymbol{v}|^{r-1}\boldsymbol{v}+\nabla p=\boldsymbol{f}(t)+``\mbox{stochastic terms}",\quad \nabla\cdot\boldsymbol{v}=0,$$ with initial and boundary vanishing conditions, where $\mu ,\alpha ,\beta >0$, $r\ge 1$ and $\mathbf{f}(\cdot )$ is a given time-dependent external force field. By the asymptotically autonomous robustness of a non-autonomous random attractor $\mathcal{A}=\{\mathcal{A}(\tau ,\omega ):\tau \in \mathbb{R},\omega \in \Omega \}$ 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) $r\in (3,\infty )$ 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 $(-\infty ,\tau ]$. This can be done by a method based on Kuratowski's measure of noncompactness.

arXiv admin note: text overlap with arXiv:2208.06808