论文标题
多孔中等方程的随机均质化
Stochastic homogenization of a porous-medium type equation
论文作者
论文摘要
我们考虑随机多孔 - 中型方程的均质化问题$ \ p_ {t} u^ε=Δf\ left（t \ left（\ frac {x} {x} {\ ep} \ ep} \ right）\ om，u^\ ep ep \ right）$，具有良好的初始基础，$ ftression $ ftestion $ ftression，$ f \ f \ f \ f（ $ u $，在给定的概率空间上$（\ om，\ mathcal {f}，μ）$ endoved en ged，带有ergodic动态系统$ \ {t（y）\，：\，y \ in \ r^n \} $。与以前的文献\ cite {afs，fs}不同，我们不假设$ \ om $ compact。我们首先表明弱解决方案$ u^\ ep $满足方程式的动力学表述，然后我们利用了理论 \ cite {bmw}开发的“均值中的两尺度收敛”，以显示动力学溶液与形式的同质化问题的动力学溶液的收敛性 $ \ p_ {t} \ +叠加{ 然后随之而来的较弱溶液的均匀化结果。
We consider the homogenization problem for the stochastic porous-medium type equation $\p_{t} u^ε=Δf\left(T\left(\frac{x}{\ep}\right)\om,u^\ep\right)$, with a well-prepared initial datum, where $f(T(y)\om,u)$ is a stationary process, increasing in $u$, on a given probability space $(\Om, \mathcal{F}, μ)$ endowed with an ergodic dynamical system $\{T(y)\,:\,y\in\R^N\}$. Differently from the previous literature \cite{afs,fs}, here we do not assume $\Om$ compact. We first show that the weak solution $u^\ep$ satisfies a kinetic formulation of the equation, then we exploit the theory of "stochastically two-scale convergence in the mean" developed in \cite{bmw} to show convergence of the kinetic solution to the kinetic solution of an homogenized problem of the form $\p_{t} \overline{u} - Δ\overline{f}(\overline{u})=0$. The homogenization result for the weak solutions then follows.
