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

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.