We are dealing with the Navier-Stokes equation in a bounded regular domain $D$ 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 $deltadownarrow 0$, so that the noise converges to the white noise in space and time. For every $delta>0$ we introduce the large deviation action functional $S^delta_{0,T}$ and the corresponding quasi-potential $V_delta$ and, by using arguments from relaxation and $Gamma$-convergence, we show that $V_delta$ converges to $V=V_0$, in spite of the fact that the Navier-Stokes equation has no meaning, when perturbed by space-time white noise. Moreover, in the case of periodic boundary conditions the limiting functional $V$ is explicitly computed.
Finally, we apply these results to estimate the asymptotics of theexpected 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. ( Slides)
Speaker: Sandra Cerrai, Department of Mathematics, University of Maryland