{
  "id": "1509.06325",
  "version": "v1",
  "published": "2015-09-21T17:51:33.000Z",
  "updated": "2015-09-21T17:51:33.000Z",
  "title": "Porous Medium Flow with both a Fractional Potential Pressure and Fractional Time Derivative",
  "authors": [
    "Mark Allen",
    "Luis Caffarelli",
    "Alexis Vasseur"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study a porous medium equation with right hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator. The derivative in time is also fractional of Caputo-type and which takes into account \"memory''. The precise model is \\[ D_t^{\\alpha} u - \\text{div}(u(-\\Delta)^{-\\sigma} u) = f, \\quad 0<\\sigma <1/2. \\] We pose the problem over $\\{t\\in {\\mathbb R}^+, x\\in {\\mathbb R}^n\\}$ with nonnegative initial data $u(0,x)\\geq 0 $ as well as right hand side $f\\geq 0$. We first prove existence for weak solutions when $f,u(0,x)$ have exponential decay at infinity. Our main result is H\\\"older continuity for such weak solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-21T17:51:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K55",
      "35K65"
    ],
    "keywords": [
      "fractional potential pressure",
      "porous medium flow",
      "fractional time derivative",
      "right hand side",
      "weak solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150906325A"
    }
  }
}