ASYMPTOTICALLY AUTONOMOUS ROBUSTNESS IN PROBABILITY OF RANDOM ATTRACTORS FOR STOCHASTIC NAVIER-STOKES EQUATIONS ON UNBOUNDED POINCARE\' DOMAINS

The asymptotically autonomous robustness of random attractors of stochastic fluid equations defined on bounded domains has been considered in the literature. In this article, we initially consider this topic (almost surely and in probability) for a nonautonomous stochastic 2D Navier-Stokes equation driven by additive and multiplicative noises defined on some unbounded Poincare'\ domains. There are two significant keys to studying this topic: (1) determining the asymptotically autonomous limiting set of the time-section of random attractors as time goes to negative infinity, and (2) showing the precompactness of a time-union of random attractors over an infinite time-interval (-oo,\tau ]. We guess and prove that such a limiting set is just determined by the random attractor of a stochastic Navier-Stokes equation (SNSE) driven by an autonomous forcing satisfying a convergent condition. The uniform ``tail-smallness"" and ``flattening effect"" of the solu-tions are derived in order to justify that the usual asymptotic compactness of the solution operators is uniform over (-oo,\tau ]. This in fact leads to the precompactness of the time-union of random attractors over (-oo,\tau ]. The idea of uniform tail-estimates due to Wang [Phys. D, 128 (1999), pp. 41--52] 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.