Ergodicity for the randomly forced Navier-Stokes system in a two-dimensional unbounded domain

TL;DR

This paper proves the uniqueness and exponential mixing of the stationary measure for the 2D Navier-Stokes system in an unbounded domain with random forcing, extending ergodic theory beyond bounded domains.

Contribution

It establishes ergodic properties for the 2D Navier-Stokes system in unbounded domains, a setting previously limited to bounded domains, using controllability and asymptotic compactness techniques.

Findings

Proves uniqueness of stationary measure

Establishes exponential mixing in dual-Lipschitz metric

Extends ergodic results to unbounded domains

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 Poincar\'e 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 arXiv:1803.01893 and arXiv:1802.03250 and using the asymptotic compactness of the dynamics.