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

Abstract

The paper is devoted to studying the asymptotics of the family $(\mu ^\varepsilon )_{\varepsilon >0}$ of stationary measures of the Markov process generated by the flow of equation

$$\begin{aligned} \dot{u}-\Delta u+(u,\nabla )u+\nabla p=h(x)+\sqrt{\varepsilon }\,\vartheta (t,x), \quad {\mathrm{div}\,}u=0, \quad u|_{\partial D}=0 \end{aligned}$$

in a bounded domain $D\subset {\mathbb {R}^2}$, where $h\in H^1_0(D)$ and $\vartheta$ is a spatially regular white noise. By using the large deviation techniques, we prove that the family $(\mu ^\varepsilon )$ is exponentially tight in $H^{1-\gamma }(D)$ for any $\gamma >0$ and vanishes exponentially outside any neighborhood of the set ${{\mathcal {O}}}$ of $\omega$-limit points of the deterministic equation. In particular, any of its weak limits is concentrated on the closure $\bar{{\mathcal {O}}}$. 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.