{
  "id": "1305.4167",
  "version": "v1",
  "published": "2013-05-17T19:35:41.000Z",
  "updated": "2013-05-17T19:35:41.000Z",
  "title": "Homogenization of a generalized Stefan Problem in the context of ergodic algebras",
  "authors": [
    "Hermano Frid",
    "Jean Silva",
    "Henrique Versieux"
  ],
  "categories": [
    "math.AP"
  ],
  "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 $u(x,t)$, $$ \\frac{\\partial}{\\partial t}\\partial_u \\Psi(x/\\ve,x,u)-\\nabla_x\\cdot \\nabla_\\eta\\psi( x/\\ve,x,t,u,\\nabla u) \\ni f(x/\\ve,x,t, u), $$ 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 $\\AA$. 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 the usual $L^2$ convergence in the cartesian product $\\Pi\\X\\R^n$, 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 $L^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-17T19:35:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B27"
    ],
    "keywords": [
      "generalized stefan problem",
      "homogenization",
      "arbitrary ergodic algebra",
      "two-scale convergence technique",
      "periodic case relies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.4167F"
    }
  }
}