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Asymptotically autonomous robustness in probability of random attractors for stochastic Navier-Stokes equations on unbounded Poincaré domains

The asymptotically autonomous robustness of random attractors of stochastic fluid equations defined on \emph{bounded} domains has been considered in the literature. In this article, we initially consider this topic (almost surely and in probability) for a non-autonomous stochastic 2D Navier-Stokes equation driven by additive and multiplicative noise defined on some \emph{unbounded Poincaré domains}. There are two significant keys to study this topic: what is the asymptotically autonomous limiting set of the time-section of random attractors as time goes to negative infinity, and how to show the precompactness of a time-union of random attractors over an \emph{infinite} time-interval $(-\infty,τ]$. We guess and prove that such a limiting set is just determined by the random attractor of a stochastic Navier-Stokes equation driven by an autonomous forcing satisfying a convergent condition. The uniform "tail-smallness" and "flattening effecting" of the solutions are derived in order to justify that the usual asymptotically compactness of the solution operators is \emph{uniform} over $(-\infty,τ]$. This in fact leads to the precompactness of the time-union of random attractors over $(-\infty,τ]$. The idea of uniform tail-estimates due to Wang \cite{UTE-Wang} is employed to overcome the noncompactness of Sobolev embeddings on unbounded domains. Several rigorous calculations are given to deal with the pressure terms when we derive these uniform tail-estimates.