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 $$\mathcal {O}$$O of $$\mathbb {R}^2$$R2, perturbed by an additive Gaussian noise $$\partial w^{Q_\delta }/\partial t$$∂wQδ/∂t, which is white in time and colored in space. We assume that the correlation radius of the noise gets smaller and smaller as $$\delta \searrow 0$$δ↘0, so that the noise converges to the white noise in space and time. For every $$\delta >0$$δ>0 we introduce the large deviation action functional $$S^\delta _{T}$$STδ and the corresponding quasi-potential $$U_\delta $$Uδ and, by using arguments from relaxation and $$\Gamma $$Γ-convergence we show that $$U_\delta $$Uδ converges to $$U=U_0$$U=U0, 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 $$U$$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.