{
  "id": "1409.8190",
  "version": "v1",
  "published": "2014-09-29T17:04:33.000Z",
  "updated": "2014-09-29T17:04:33.000Z",
  "title": "Regularity of solutions of the fractional porous medium flow with exponent 1/2",
  "authors": [
    "Luis Caffarelli",
    "Juan Luis Vázquez"
  ],
  "comment": "35 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:1201.6048",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the regularity of a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is $u_t=\\nabla\\cdot(u\\nabla (-\\Delta)^{-1/2}u).$ For definiteness, the problem is posed in $\\{x\\in\\mathbb{R}^N, t\\in \\mathbb{R}\\}$ with nonnegative initial data $u(x,0)$ that are integrable and decay at infinity. Previous papers have established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation, as well as the boundedness of nonnegative solutions with $L^1$ data, for the more general family of equations $u_t=\\nabla\\cdot(u\\nabla (-\\Delta)^{-s}u)$, $0<s<1$. Here we establish the $C^\\alpha$ regularity of such weak solutions in the difficult fractional exponent case $s=1/2$. For the other fractional exponents $s\\in (0,1)$ this H\\\"older regularity has been proved in $[5]$. The method combines delicate De Giorgi type estimates with iterated geometric corrections that are needed to avoid the divergence of some essential energy integrals due to fractional long-range effects.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-29T17:04:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K55",
      "35K65",
      "76S05"
    ],
    "keywords": [
      "fractional porous medium flow",
      "regularity",
      "weak solutions satisfying energy estimates",
      "fractional exponent",
      "inverse fractional laplacian operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.8190C"
    }
  }
}