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Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains. (English) Zbl 1113.60062
This paper is devoted to study the behaviour of stochastic partial differential equations in unbounded domains. The authors begin by considering random dynamical systems (RDS) on a separable Banach space. They introduce the new concept of asymptotically compact (AC) RDS. They prove that for an ACRDS, the ${\Omega }$-limit set of any bounded subset $B$ is nonempty, compact, invariant and attracts $B$. The invariance of this set will imply the existence of invariant measure, while the uniqueness is not considered. Using the classical Galerkin approximation method and some compactness theorem they prove the existence of the stochastic flow associated with 2D stochastic Navier-Stokes equations in Poincaré domains (possibly unbounded). Then, they construct the RDS corresponding to the Navier-Stokes equations. Using energy inequalities they are able to prove the continuity of this RDS in a weak topology and that this RDS is AC. Finally, they obtain the existence of an invariant measure for the 2D stochastic Navier-Stokes equations perturbed by an additive noise. Since the method used by the authors does not depend on the compactness of the Sobolev embeddings, they can deal with unbounded domains and they can relax assumptions on the noise. So, they obtain new results in bounded and in unbounded domains.

##### MSC:
 60H15 Stochastic partial differential equations 35R60 PDEs with randomness, stochastic PDE 37H10 Generation, random and stochastic difference and differential equations 34F05 ODE with randomness