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.
Highlights
This paper continues our study of the interconnection between controllability and mixing properties of random dynamical systems
In the first part of this project [Shi17a], we studied a class of ordinary differential equations driven by vector fields with random amplitudes and proved that good knowledge of controllability properties ensures the uniqueness of a stationary distribution and exponential convergence to it in the total variation metric
The additional properties of the resolving operator that are established will be important when proving the exponential mixing of the random flow associated with the 2D Navier–Stokes system
Summary
In the first part of this project [Shi17a], we studied a class of ordinary differential equations driven by vector fields with random amplitudes and proved that good knowledge of controllability properties ensures the uniqueness of a stationary distribution and exponential convergence to it in the total variation metric. Let us mention that this paper is a part of the programme whose goal is to develop methods for applying the results and tools of the control theory in the study of mixing properties of flows generated by randomly forced evolution equations. It complements the earlier results established in [AKSS07, Shi[15], Shi17a, KNS18] and develops a general framework for dealing with random perturbations acting through the boundary of the domain. The appendix gathers a few auxiliary results used in the main text
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