Controllability implies mixing II. Convergence in the dual-Lipschitz metric

TL;DR

This paper establishes that approximate controllability and local stabilization lead to unique stationary measures and exponential mixing in the dual-Lipschitz metric, with applications to boundary-driven 2D Navier-Stokes equations.

Contribution

It provides an abstract framework linking controllability and mixing, and applies it to boundary-driven Navier-Stokes systems, demonstrating local exponential stabilization.

Findings

Unique stationary measure under controllability assumptions

Exponential mixing in the dual-Lipschitz metric

Local exponential stabilization of boundary-driven Navier-Stokes

Abstract

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 by-product of our analysis is the local exponential stabilisation of the boundary-driven Navier-Stokes system by a regular boundary control.