{
  "id": "1902.10187",
  "version": "v1",
  "published": "2019-02-26T20:00:18.000Z",
  "updated": "2019-02-26T20:00:18.000Z",
  "title": "Numerical Approximation of Young Measure Solutions to Parabolic Systems of Forward-Backward Type",
  "authors": [
    "Miles Caddick",
    "Endre Süli"
  ],
  "comment": "31 pages",
  "categories": [
    "math.NA"
  ],
  "abstract": "This paper is concerned with the proof of existence and numerical approximation of large-data global-in-time Young measure solutions to initial-boundary-value problems for multidimensional nonlinear parabolic systems of forward-backward type of the form $\\partial_t u - \\mbox{div}(a(Du)) + Bu = F$, where $B \\in \\mathbb{R}^{m \\times m}$, $Bv \\cdot v \\geq 0$ for all $v \\in \\mathbb{R}^m$, $F$ is an $m$-component vector-function defined on a bounded open Lipschitz domain $\\Omega \\subset \\mathbb{R}^n$, and $a$ is a locally Lipschitz mapping of the form $a(A)=K(A)A$, where $K\\,:\\, \\mathbb{R}^{m \\times n} \\rightarrow \\mathbb{R}$. The function $a$ may have a nonstandard growth rate, in the sense that it is permitted to have unequal lower and upper growth rates. Furthermore, $a$ is not assumed to be monotone, nor is it assumed to be the gradient of a potential. Problems of this type arise in mathematical models of the atmospheric boundary layer and fall beyond the scope of monotone operator theory. We develop a numerical algorithm for the approximate solution of problems in this class, and we prove the convergence of the algorithm to a Young measure solution of the system under consideration.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-26T20:00:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K51",
      "35K55",
      "76M10"
    ],
    "keywords": [
      "numerical approximation",
      "forward-backward type",
      "large-data global-in-time young measure solutions",
      "growth rate",
      "multidimensional nonlinear parabolic systems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}