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),∂∂t∂uΨ(x/ε,x,u)−∇x⋅∇ηψ(x/ε,x,t,u,∇u)∋f(x/ε,x,t,u), on a bounded domain Ω⊆Rn, t∈(0,T), together with initial–boundary conditions, where Ψ(z,x,⋅) is strictly convex and ψ(z,x,t,u,⋅) is a C1 convex function, both with quadratic growth, satisfying some additional technical hypotheses. As functions of the oscillatory variable, Ψ(⋅,x,u),ψ(⋅,x,t,u,η) and f(⋅,x,t,u) belong to the generalized Besicovitch space B2 associated with an arbitrary ergodic algebra 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 L2 convergence in the Cartesian product Π×Rn, where Π 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 L2.