Large deviations for invariant measures of white-forced 2D Navier-Stokes equation

TL;DR

This paper investigates the asymptotic behavior of stationary measures for the stochastic 2D Navier-Stokes equations with white noise, establishing large deviations principles and exponential tightness in certain function spaces.

Contribution

It introduces a new formula for recovering stationary measures of Markov processes with good mixing properties using local information, and applies large deviations techniques to stochastic Navier-Stokes equations.

Findings

Family of stationary measures is exponentially tight in $H^{1- ext{gamma}}(D)$.

Measures vanish exponentially outside neighborhoods of deterministic omega-limit points.

In trivial dynamics, measures satisfy the large deviations principle.

Abstract

The paper is devoted to studying the asymptotics of the family $(\mu^\varepsilon)$ of stationary measures of the Markov process generated by the flow of stochastic 2D Navier-Stokes equation with smooth white noise. By using the large deviations techniques, we prove that this family is exponentially tight in $H^{1-\gamma}(D)$ for any $\gamma>0$ and vanishes exponentially outside any neighborhood of the set $\cal O$ of $\omega$-limit points of the deterministic equation. In particular, any of its weak limits is concentrated on the closure $\bar{\cal O}$. A key ingredient of the proof is a new formula that allows to recover the stationary measure $\mu$ of a Markov process with good mixing properties, knowing only some local information about $\mu$. In the case of trivial limiting dynamics, our result implies that the family $(\mu^\varepsilon)$ obeys the large deviations principle.