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We are dealing with the Navier-Stokes equation in a bounded regular domain$D$ of $\mathbb{R}^2$, perturbed by an additive Gaussian noise $\partialw^{Q_\delta}/\partial t$, which is white in time and colored in space. Weassume that the correlation radius of the noise gets smaller and smaller as$\delta\searrow 0$, so that the noise converges to the white noise in space andtime. For every $\delta>0$ we introduce the large deviation action functional$S^\delta_{0,T}$ and the corresponding quasi-potential $U_\delta$ and, by usingarguments from relaxation and $\Gamma$-convergence we show that $U_\delta$converges to $U=U_0$, in spite of the fact that the Navier-Stokes equation hasno meaning in the space of square integrable functions, when perturbed byspace-time white noise. Moreover, in the case of periodic boundary conditionsthe limiting functional $U$ is explicitly computed.  Finally, we apply these results to estimate of the asymptotics of theexpected exit time of the solution of the stochastic Navier-Stokes equationfrom a basin of attraction of an asymptotically stable point for theunperturbed system.

Quasipotential and exit time for 2D Stochastic Navier-Stokes equations driven by space time white noise