时空白噪声驱动下二维随机Navier-斯托克斯方程的拟势和出口时间
Quasipotential and exit time for 2D Stochastic Navier-Stokes equations driven by space time white noise
摘要：
我们正在处理的Navier-Stokes方程的有界正则区域$D$的$\mathbb{R}^2$，扰动的加性高斯噪声$\partial w^{Q_δ}/\partial t$，这是白色的时间和空间的颜色。我们假设噪声的相关半径随着$δ\searrow 0$的减小而减小，从而噪声在空间和时间上收敛到白噪声。对于每一个$δ>0$，我们引入大偏差作用泛函$S^δ_{0,T}$和相应的准势$U_δ$，并通过使用从松弛和$Γ$ -收敛的参数，我们表明，$U_δ$收敛到$U=U_0$，尽管事实上，Navier-Stokes方程在平方可积函数的空间中没有意义，当扰动时空白噪声。此外，在周期性边界条件的情况下，明确计算的极限功能$U$。最后，我们应用这些结果估计的期望退出时间的随机Navier-Stokes方程的解从一个吸引盆的渐近稳定点的未扰动系统。
英文摘要：
We are dealing with the Navier-Stokes equation in a bounded regular domain $D$ of $\mathbb{R}^2$, perturbed by an additive Gaussian noise $\partial w^{Q_δ}/\partial t$, which is white in time and colored in space. We assume that the correlation radius of the noise gets smaller and smaller as $δ\searrow 0$, so that the noise converges to the white noise in space and time. For every $δ>0$ we introduce the large deviation action functional $S^δ_{0,T}$ and the corresponding quasi-potential $U_δ$ and, by using arguments from relaxation and $Γ$-convergence we show that $U_δ$ converges to $U=U_0$, 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 $U$ 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.
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