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

TY - JOUR

T1 - Quasipotential and exit time for 2D Stochastic Navier-Stokes equations driven by space time white noise

AU - Brzezniak, Z.

AU - Cerrai, S.

AU - Freidlin, M.

PY - 2015/8

Y1 - 2015/8

N2 - 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.

AB - 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.

KW - 60H15

KW - 60F10

KW - 35Q30

KW - 49J45

UR - http://www.scopus.com/inward/record.url?scp=84911900355&partnerID=8YFLogxK

U2 - 10.1007/s00440-014-0584-6

DO - 10.1007/s00440-014-0584-6

M3 - Article

SN - 0178-8051

VL - 162

SP - 739

EP - 793

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

ER -