{
  "id": "1007.3744",
  "version": "v1",
  "published": "2010-07-21T20:04:50.000Z",
  "updated": "2010-07-21T20:04:50.000Z",
  "title": "On the global existence for the Muskat problem",
  "authors": [
    "Peter Constantin",
    "Diego Cordoba",
    "Francisco Gancedo",
    "Robert M. Strain"
  ],
  "comment": "31 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an $L^2(\\R)$ maximum principle, in the form of a new ``log'' conservation law \\eqref{ln} which is satisfied by the equation \\eqref{ec1d} for the interface. Our second result is a proof of global existence of Lipschitz continuous solutions for initial data that satisfy $\\|f_0\\|_{L^\\infty}<\\infty$ and $\\|\\partial_x f_0\\|_{L^\\infty}<1$. We take advantage of the fact that the bound $\\|\\partial_x f_0\\|_{L^\\infty}<1$ is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law. Lastly, we prove a global existence result for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance $\\| f\\|_1 \\le 1/5$. Previous results of this sort used a small constant $\\epsilon \\ll1$ which was not explicit.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-07-21T20:04:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "grants strong compactness properties",
      "initial data",
      "muskat problem models",
      "unique strong solutions",
      "global existence result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.3744C"
    }
  }
}