{
  "id": "1804.10403",
  "version": "v1",
  "published": "2018-04-27T09:00:42.000Z",
  "updated": "2018-04-27T09:00:42.000Z",
  "title": "Well-posedness and stability results for some periodic Muskat problems",
  "authors": [
    "Bogdan-Vasile Matioc"
  ],
  "comment": "52 pages, 1 figure",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the two-dimensional Muskat problem in a horizontally periodic setting and for fluids with arbitrary densities and viscosities. We show that in the presence of surface tension effects the Muskat problem is a quasilinear parabolic problem which is well-posed in the Sobolev space $H^r(\\mathbb{S})$ for each $r\\in(2,3)$. When neglecting surface tension effects, the Muskat problem is a fully nonlinear evolution equation and of parabolic type in the regime where the Rayleigh-Taylor condition is satisfied. We then establish the well-posedness of the Muskat problem in the open subset of $H^2(\\mathbb{S})$ defined by the Rayleigh-Taylor condition. Besides, we identify all equilibrium solutions and study the stability properties of trivial and of small finger-shaped equilibria. Also other qualitative properties of solutions such as parabolic smoothing, blow-up behavior, and criteria for global existence are outlined.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-27T09:00:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B35",
      "35B65",
      "35K55",
      "35Q35",
      "42B20"
    ],
    "keywords": [
      "periodic muskat problems",
      "stability results",
      "well-posedness",
      "rayleigh-taylor condition",
      "quasilinear parabolic problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}