{
  "id": "1610.05546",
  "version": "v1",
  "published": "2016-10-18T11:41:41.000Z",
  "updated": "2016-10-18T11:41:41.000Z",
  "title": "The Muskat problem in 2D: equivalence of formulations, well-posedness, and regularity results",
  "authors": [
    "Bogdan-Vasile Matioc"
  ],
  "comment": "40 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we consider the Muskat problem describing the motion of two unbounded immiscible fluid layers with equal viscosities in vertical or horizontal two-dimensional geometries. We first prove that the mathematical model can be formulated as an evolution problem for the sharp interface separating the two fluids, which turns out to be, in a suitable functional analytic setting, quasilinear and of parabolic type. Based upon these properties, we then establish the local well-posedness of the problem for arbitrary large initial data and show that the solutions become instantly real-analytic in time and space. Our method allows us to choose the initial data in the class $H^s,$ $s\\in(3/2,2)$, when neglecting surface tension, respectively in $H^s,$ $s\\in(2,3),$ when surface tension effects are included. Besides, we provide new criteria for the global existence of solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-18T11:41:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R37",
      "35K59",
      "35K93",
      "35Q35",
      "42B20"
    ],
    "keywords": [
      "muskat problem",
      "regularity results",
      "well-posedness",
      "formulations",
      "equivalence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}