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

We are dealing with the Navier-Stokes equation in a bounded regular domain O of R 2 , perturbed by an additive Gaussian noise ∂ w Q δ / ∂ t , which is white in time and colored in space. We assume that the correlation radius of the noise gets smaller and smaller as δ ↘ 0 , so that the noise converges to the white noise in space and time. For every δ > 0 we introduce the large deviation action functional S T δ and the corresponding quasi-potential U δ and, by using arguments from relaxation and Γ -convergence we show that U δ converges to U = U 0 , in spite of the fact that the Navier-Stokes equation has no meaning in the space of square integrable functions, when perturbed by space-time white noise. Moreover, in the case of periodic boundary conditions the limiting functional U is explicitly computed. Finally, we apply these results to estimate of the asymptotics of the expected exit time of the solution of the stochastic Navier-Stokes equation from a basin of attraction of an asymptotically stable point for the unperturbed system.