Ergodicity for the Randomly Forced Navier–Stokes System in a Two-Dimensional Unbounded Domain

Abstract

The ergodic properties of the randomly forced Navier–Stokes system have been extensively studied in the literature during the last two decades. The problem has always been considered in bounded domains, in order to have, for example, suitable spectral properties for the Stokes operator, to ensure some compactness properties for the resolving operator of the system and the associated functional spaces, etc. In the present paper, we consider the Navier–Stokes system in an unbounded domain satisfying the Poincaré inequality. Assuming that the system is perturbed by a bounded non-degenerate noise, we establish uniqueness of stationary measure and exponential mixing in the dual-Lipschitz metric. The proof is carried out by developing the controllability approach of the papers Kuksin et al. (Geom Funct Anal 30(1):126–187, 2020) and Shirikyan (J Eur Math Soc, 2020) and using the asymptotic compactness of the dynamics.