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

Abstract

We are dealing with the Navier-Stokes equation in a bounded regular domain $$\mathcal {O}$$ of $$\mathbb {R}^2$$ , perturbed by an additive Gaussian noise $$\partial w^{Q_\delta }/\partial t$$ , which is white in time and colored in space. We assume that the correlation radius of the noise gets smaller and smaller as $$\delta \searrow 0$$ , so that the noise converges to the white noise in space and time. For every $$\delta >0$$ we introduce the large deviation action functional $$S^\delta _{T}$$ and the corresponding quasi-potential $$U_\delta $$ and, by using arguments 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 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.