**SMS scnews item created by Daniel Daners at Tue 2 Sep 2014 1648**

Type: Seminar

Modified: Wed 3 Sep 2014 0910

Distribution: World

Expiry: 8 Sep 2014

**Calendar1: 8 Sep 2014 1400-1500**

**CalLoc1: AGR Carslaw 829**

Auth: daners@como.maths.usyd.edu.au

### PDE Seminar

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

### Brzezniak

Zdzisław Brzeźniak

University of York, UK

8 Sep 2014, 2-3pm, Carslaw Room 829 (AGR)

## Abstract

We are dealing with the Navier-Stokes equation in a bounded regular domain
$D$ of
${\mathbb{R}}^{2}$, perturbed by an
additive Gaussian noise $\partial {w}^{\delta}\u2215\partial t$,
that 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
${S}_{0,T}^{\delta}$ and the corresponding
quasi-potential ${U}_{\delta}$
and, by using arguments from relaxation and
$\Gamma $-convergence we show
that ${U}_{\delta}$ converges to a
limiting quasi-potential ${U}_{0}$,
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
${U}_{0}$ is
explicitly computed.

Finally, we apply these results to estimate 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.

This talk is based on a joint work with Sandra Cerrai and Mark Freidlin.

Check also the PDE
Seminar page. Enquiries to Daniel Hauer or Daniel Daners.