Quasipotential and exit time for 2D Stochastic Navier-Stokes equations driven by space time white noise
Quasipotential and exit time for 2D Stochastic Navier-Stokes equations driven by space time white noise
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DOI：
10.1007/s00440-014-0584-6
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发表时间：
2014-01
影响因子：
2
通讯作者：
Zdzisław Brzeźniak;S. Cerrai;M. Freidlin
中科院分区：
文献类型：
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作者：
Zdzisław Brzeźniak;S. Cerrai;M. Freidlin
我们正在有界正则域中处理纳维-斯托克斯方程，该方程受到加性高斯噪声的扰动，该噪声在时间上是白色的，在空间上是彩色的。我们假设噪声的相关半径越来越小，使得噪声在空间和时间上收敛到白噪声。对于每一个我们引入的大偏差作用泛函和相应的拟势和，通过使用松弛和收敛的论证，我们表明，尽管纳维-斯托克斯方程在平方可积函数空间中没有意义，但当受到时空白噪声扰动时，收敛到 。此外，在周期性边界条件的情况下，显式计算限制泛函。最后，我们将这些结果应用于从未扰动系统的渐近稳定点的吸引盆中估计随机纳维-斯托克斯方程解的预期退出时间的渐近。
We are dealing with the Navier-Stokes equation in a bounded regular domainof, perturbed by an additive Gaussian noise, which is white in time and colored in space. We assume that the correlation radius of the noise gets smaller and smaller as, so that the noise converges to the white noise in space and time. For everywe introduce the large deviation action functionaland the corresponding quasi-potentialand, by using arguments from relaxation and-convergence we show thatconverges to, 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 functionalis 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.