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

# 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 ${ℝ}^{2}$, perturbed by an additive Gaussian noise $\partial {w}^{\delta }∕\partial t$, that is white in time and colored in space. We assume that the correlation radius of the noise gets smaller and smaller as $\delta ↘0$, 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.

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