Periodic Random Attractors for Stochastic Navier-Stokes Equations on Unbounded Domains

This paper is concerned with the asymptotic behavior of solutions of the two-dimensional Navier-Stokes equations with both non-autonomous deterministic and stochastic terms defined on unbounded domains. We first introduce a continuous cocycle for the equations and then prove the existence and uniqueness of tempered random attractors. We also characterize the structures of the random attractors by complete solutions. When deterministic forcing terms are periodic, we show that the tempered random attractors are also periodic. Since the Sobolev embeddings on unbounded domains are not compact, we establish the pullback asymptotic compactness of solutions by Ball's idea of energy equations.