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

PDE Seminar
Tuesday, February 19, 2013 - 3:00pm
1 hour (actually 50 minutes)
Skiles 006
IMPA, Rio De Janeiro, Braizil
We address the deterministic homogenization, in the general context of ergodic algebras, of a doubly nonlinear problem whichgeneralizes 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 $u(x,t)$, $$0\in \frac{\partial}{\partial t}\partial_u \Psi(x/\ve,x,u)+\nabla_x\cdot \nabla_\eta\psi(x/\ve,x,t,u,\nabla u) - f(x/\ve,x,t, u), $$ 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 $\AA$. 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 usual $L^2$ convergence in the Cartesian product $\Om\X\Pi$, 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 uniformly bounded  sequences in $L^2$.