Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains

Abstract

Abstract. We introduce a notion of an asymptotically compact (AC) random dynamical system (RDS). We prove that for an AC RDS the Ω-limit set Ω B (ω) of any bounded set B is nonempty, compact, strictly invariant and attracts the set B. We establish that the 2D Navier Stokes Equations (NSEs) in a domain satisfying the Poincaré inequality perturbed by an additive irregular noise generate an AC RDS in the energy space H. As a consequence we deduce existence of an invariant measure for such NSEs. Our study generalizes on the one hand the earlier results by Flandoli-Crauel (1994) and Schmalfuss (1992) obtained in the case of bounded domains and regular noise, and on the other hand the results by Rosa (1998) for the deterministic NSEs.

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