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

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JournalProbability Theory and Related Fields
DateE-pub ahead of print - 18 Nov 2014
DatePublished (current) - Aug 2015
Volume162
Number of pages55
Pages (from-to)739–793
Early online date18/11/14
Original languageEnglish

Abstract

We are dealing with the Navier-Stokes equation in a bounded regular domain O of R2, perturbed by an additive Gaussian noise ∂wQδ/∂t, which is white in time and colored in space. We assume that the correlation radius of the noise gets smaller and smaller as δ↘0, so that the noise converges to the white noise in space and time. For every δ>0 we introduce the large deviation action functional SδT and the corresponding quasi-potential Uδ and, by using arguments from relaxation and Γ-convergence we show that Uδ converges to U=U0, 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.

    Research areas

  • 60H15, 60F10, 35Q30, 49J45

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