Polynomial mixing for the white-forced Navier-Stokes system in the whole space

TL;DR

This paper investigates the mixing behavior of the two-dimensional Navier-Stokes equations driven by white noise in an unbounded domain, establishing uniqueness of the stationary measure and polynomial mixing rates under certain noise conditions.

Contribution

It introduces a novel combination of coupling, Foiaș-Prodi estimates, and weighted analysis to prove polynomial mixing for the stochastic Navier-Stokes system in the whole space.

Findings

Proves uniqueness of stationary measure for the system.

Establishes polynomial mixing rates in the dual-Lipschitz metric.

Develops new weighted estimates involving Muckenhoupt weights.

Abstract

We study the mixing properties of the white-forced Navier-Stokes system in the whole space $\mathbb{R}^2$. Assuming that the noise is sufficiently non-degenerate, we prove the uniqueness of stationary measure and polynomial mixing in the dual-Lipschitz metric. The proof combines the coupling method with a Foia\c{s}-Prodi type estimate, weighted growth estimates for trajectories, and an estimate for the Leray projector involving Muckenhoupt $A_2$-class weights.