Asymptotic Autonomy of Random Attractors in Regular Spaces for Non-autonomous Stochastic Navier-Stokes Equations

TL;DR

This paper proves the existence of unique pullback attractors for non-autonomous stochastic Navier-Stokes equations, demonstrating their asymptotic autonomy in regular function spaces under multiplicative and additive noise.

Contribution

It establishes the asymptotic autonomy of random attractors in regular spaces for non-autonomous stochastic Navier-Stokes equations, using backward-uniform flattening and pullback asymptotic compactness.

Findings

Existence of unique pullback attractors in regular spaces.

Attractors are backward compact and asymptotically autonomous.

Use of backward-uniform flattening property for compactness.

Abstract

This article concerns the long-term random dynamics in regular spaces for a non-autonomous Navier-Stokes equation defined on a bounded smooth domain $\mathcal{O}$ driven by multiplicative and additive noise. For the two kinds of noise driven equations, we demonstrate the existence of a unique pullback attractor which is backward compact and asymptotically autonomous in $\mathbb{L}^2(\mathcal{O})$ and $\mathbb{H}_0^1(\mathcal{O})$, respectively. The backward-uniform flattening property of the solution is used to prove the backward-uniform pullback asymptotic compactness of the non-autonomous random dynamical systems in the regular space $\mathbb{H}_0^1(\mathcal{O})$.