Asymptotically autonomous robustness in probability of random attractors for stochastic Navier-Stokes equations on unbounded Poincar\'e domains

TL;DR

This paper investigates the robustness of random attractors for stochastic 2D Navier-Stokes equations on unbounded domains, establishing their asymptotic behavior and precompactness over infinite time intervals.

Contribution

It extends the analysis of asymptotically autonomous robustness of random attractors from bounded to unbounded domains, introducing new techniques for uniform tail-estimates and handling pressure terms.

Findings

The limiting set of random attractors is determined by autonomous forcing conditions.

Precompactness of the union of random attractors over (-∞, τ] is established.

Uniform tail-estimates are successfully applied on unbounded domains.

Abstract

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 Poincar\'e 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 $(-\infty,\tau]$. 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 $(-\infty,\tau]$. This in fact leads to the precompactness of the time-union of random attractors over $(-\infty,\tau]$. 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.