Homogenization of a generalized Stefan Problem in the context of ergodic algebras

Abstract

We address the deterministic homogenization, in the general context of ergodic algebras, of a doubly nonlinear problem which generalizes the well known Stefan model, and includes the classical porous medium equation. It may be represented by the differential inclusion, for a real-valued function $u(x,t)$, $$ \frac{\partial}{\partial t}\partial_u Ψ(x/\ve,x,u)-\nabla_x\cdot \nabla_ηψ( x/\ve,x,t,u,\nabla u) \ni f(x/\ve,x,t, u), $$ on a bounded domain $\Om\subset \R^n$, $t\in (0,T)$, together with initial-boundary conditions, where $\Psi (z,x,\cdot )$ is strictly convex and $\psi (z,x,t,u,\cdot )$ is a $C^1$ convex function, both with quadratic growth, satisfying some additional technical hypotheses. As functions of the oscillatory variable, $\Psi (\cdot ,x,u),\psi (\cdot ,x,t,u,\eta )$ and $f(\cdot ,x,t,u)$ belong to the generalized Besicovitch space $\BB^2$ associated with an arbitrary ergodic algebra $\mathring{A}$. The periodic case was addressed by Visintin (2007), based on the two-scale convergence technique. Visintin's analysis for the periodic case relies heavily on the possibility of reducing two-scale convergence to the usual $L^2$ convergence in the cartesian product $Π\X\R^n$, where $\Pi$ is the periodic cell. This reduction is no longer possible in the case of a general ergodic algebra. To overcome this difficulty, we make essential use of the concept of two-scale Young measures for algebras with mean value, associated with bounded sequences in $L^2$.