2014/01/24 by Zdzisław Brzeźniak, Sandra Cerrai, Brzezniak, Zdzislaw +3 · 2 citations
Economics, Econometrics and Finance · Mathematics · Computer Science · #Stochastic processes and financial applications #Navier-Stokes equation solutions #Advanced Mathematical Modeling in Engineering
paper · pdf · doi:10.48550/arxiv.1401.6299
We are dealing with the Navier-Stokes equation in a bounded regular domain\nD of \ℝ2, perturbed by an additive Gaussian noise \∂\nwQ_\δ/\∂ t, which is white in time and colored in space. We\nassume that the correlation radius of the noise gets smaller and smaller as\n\δ searrow 0, so that the noise converges to the white noise in space and\ntime. For every \δ>0 we introduce the large deviation action functional\nS^\δ0,T and the corresponding quasi-potential U_\δ and, by using\narguments from relaxation and \Γ-convergence we show that U_\δ\nconverges to U=U0, in spite of the fact that the Navier-Stokes equation has\nno meaning in the space of square integrable functions, when perturbed by\nspace-time white noise. Moreover, in the case of periodic boundary conditions\nthe limiting functional U is explicitly computed.\n Finally, we apply these results to estimate of the asymptotics of the\nexpected exit time of the solution of the stochastic Navier-Stokes equation\nfrom a basin of attraction of an asymptotically stable point for the\nunperturbed system.\n