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 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 L2(O) and H0(O), respectively. The compact embedding H0(O) subset of L2(O) helps us to show the backward-uniform pullback asymptotic compactness (BUPAC) of the non-autonomous random dynamical systems (NRDS) in the Lebesgue space L2(O). The backward-uniform flattening property of the solutions is used to prove the BUPAC of the NRDS in the Sobolev space H0(O).