Existence and asymptotic autonomous robustness of random attractors for three-dimensional stochastic globally modified Navier-Stokes equations on unbounded domains

TL;DR

This paper proves the existence and robustness of random attractors for 3D stochastic modified Navier-Stokes equations on unbounded domains, considering time-dependent forcing and noise perturbations, using advanced compactness and tail estimates.

Contribution

It is the first study to establish the existence and asymptotic autonomous robustness of random attractors for 3D stochastic globally modified Navier-Stokes equations on unbounded domains.

Findings

Existence of random attractors for 3D SGMNSE on unbounded domains.

Asymptotic autonomous robustness of these attractors under perturbations.

Uniform pullback asymptotic compactness across infinite time intervals.

Abstract

In this article, we discuss the existence and asymptotically autonomous robustness (AAR) (almost surely) of random attractors for 3D stochastic globally modified Navier-Stokes equations (SGMNSE) on Poincar\'e domains (which may be bounded or unbounded). Our aim is to investigate the existence and AAR of random attractors for 3D SGMNSE when the time-dependent forcing converges to a time-independent function under the perturbation of linear multiplicative noise as well as additive noise. The main approach is to provide a way to justify that, on some uniformly tempered universe, the usual pullback asymptotic compactness of the solution operators is uniform across an infinite time-interval $(-\infty,\tau]$. The backward uniform ``tail-smallness'' and ``flattening-property'' of the solutions over $(-\infty,\tau]$ have been demonstrated to achieve this goal. To the best of our knowledge, this is the first attempt to establish the existence as well as AAR of random attractors for 3D SGMNSE on unbounded domains.