{
  "id": "1908.07138",
  "version": "v1",
  "published": "2019-08-20T03:03:37.000Z",
  "updated": "2019-08-20T03:03:37.000Z",
  "title": "Well-posedness of the Fractional Porous Medium Equation on Manifolds with Conical Singularities",
  "authors": [
    "Nikolaos Roidos",
    "Yuanzhen Shao"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this article, we consider the fractional porous medium equation, $\\partial_t u +(-\\Delta)^\\sigma (|u|^{m-1}u )=0 $, posed on a Riemannian manifold with isolated conical singularities, with $m>0$ and $\\sigma\\in (0,1]$. For $L_\\infty-$initial data, we establish existence and uniqueness of a global weak solution for all $m>0$, and we show that this solution is strong for $m \\geq 1$. We further investigate a number of properties of the solutions, including comparison principle, $L_p-$contraction and conservation of mass. In particular, it is shown that mass is conserved for all $t\\geq 0$ and $m>0$. The method in this paper is quite general and thus is applicable to a variety of similar problems on manifolds with more general singularities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-20T03:03:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional porous medium equation",
      "conical singularities",
      "well-posedness",
      "global weak solution",
      "general singularities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}