{
  "id": "1612.08195",
  "version": "v1",
  "published": "2016-12-24T15:44:54.000Z",
  "updated": "2016-12-24T15:44:54.000Z",
  "title": "Well-posedness theory for degenerate parabolic equations on Riemannian manifolds",
  "authors": [
    "Melanie Graf",
    "Michael Kunzinger",
    "Darko Mitrovic"
  ],
  "comment": "32 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the degenerate parabolic equation $$ \\partial_t u +\\mathrm{div} {\\mathfrak f}_{\\bf x}(u)=\\mathrm{div}(\\mathrm{div} ( A_{\\bf x}(u) ) ), \\ \\ {\\bf x} \\in M, \\ \\ t\\geq 0 $$ on a smooth, compact, $d$-dimensional Riemannian manifold $(M,g)$. Here, for each $u\\in {\\mathbb R}$, ${\\bf x}\\mapsto {\\mathfrak f}_{\\bf x}(u)$ is a vector field and ${\\bf x}\\mapsto A_{\\bf x}(u)$ is a $(1,1)$-tensor field on $M$ such that $u\\mapsto \\langle A_{\\bf x}(u) {\\boldsymbol \\xi},{\\boldsymbol \\xi} \\rangle$, ${\\boldsymbol \\xi}\\in T_{\\bf x} M$, is non-decreasing with respect to $u$. The fact that the notion of divergence appearing in the equation depends on the metric $g$ requires revisiting the standard entropy admissibility concept. We derive it under an additional geometry compatibility condition and, as a corollary, we introduce the kinetic formulation of the equation on the manifold. Using this concept, we prove well-posedness of the corresponding Cauchy problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-24T15:44:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K65",
      "42B37",
      "76S99"
    ],
    "keywords": [
      "degenerate parabolic equation",
      "well-posedness theory",
      "additional geometry compatibility condition",
      "standard entropy admissibility concept",
      "dimensional riemannian manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}