This paper continues our study of the interconnection between controllability and mixing properties of random dynamical systems. We begin with an abstract result showing that the approximate controllability to a point and a local stabilisation property imply the uniqueness of a stationary measure and exponential mixing in the dual-Lipschitz metric. This result is then applied to the 2D Navier–Stokes system driven by a random force acting through the boundary. A byproduct of our analysis is the local exponential stabilisation of the boundary-driven Navier–Stokes system by a regular boundary control.
Cite this article
Armen Shirikyan, Controllability implies mixing II. Convergence in the dual-Lipschitz metric. J. Eur. Math. Soc. 23 (2021), no. 4, pp. 1381–1422DOI 10.4171/JEMS/1036