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

الأصل · EN

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∂ᵤ Ψ(x/,x,u)-∇ₓ· ∇ηψ(x/,x,t,u,∇ u) f(x/,x,t, u), on a bounded domain ⊂ ⁿ, 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 ² associated with an arbitrary ergodic algebra Å. 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 Πⁿ, 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.