| Contents |
| 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 $\delta\searrow 0$, so that the
noise converges to the white noise in space and time. For every
$\delta>0$ we introduce the large deviation action functional
$S^\delta_{0,T}$ 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. |
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