Asymptotic autonomy of random attractors for non-autonomous stochastic Navier-Stokes equations on bounded domains

This article concerns the long-term random dynamics 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 that 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 compact embedding $ {\mathbb{H}}_0^1(\mathcal{O})\subset{\mathbb{L}}^2(\mathcal{O}) $ helps us to show the backward-uniform pullback asymptotic compactness (BUPAC) of the non-autonomous random dynamical systems (NRDS) in the Lebesgue space $ {\mathbb{L}}^2(\mathcal{O}) $. The backward-uniform flattening property of the solutions is used to prove the BUPAC of the NRDS in the Sobolev space $ \mathbb{H}_0^1(\mathcal{O}) $.