Stochastic homogenization of a porous-medium type equation

TL;DR

This paper studies the stochastic homogenization of a porous-medium type equation with non-compact probability space, proving convergence of solutions to a deterministic homogenized equation using kinetic formulation and two-scale convergence techniques.

Contribution

It extends stochastic homogenization results to non-compact probability spaces for porous-medium equations, employing kinetic formulation and stochastic two-scale convergence methods.

Findings

Weak solutions satisfy a kinetic formulation.

Solutions converge to a homogenized PDE.

Homogenized equation is deterministic.

Abstract

We consider the homogenization problem for the stochastic porous-medium type equation $\p_{t} u^\epsilon =\Delta f\left(T\left(\frac{x}{\ep}\right)\om,u^\ep\right)$, with a well-prepared initial datum, where $f(T(y)\om,u)$ is a stationary process, increasing in $u$, on a given probability space $(\Om, \mathcal{F}, \mu)$ 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 $\p_{t} \overline{u} - \Delta \overline{f}(\overline{u})=0$. The homogenization result for the weak solutions then follows.