{
  "id": "1706.09260",
  "version": "v1",
  "published": "2017-06-28T12:50:35.000Z",
  "updated": "2017-06-28T12:50:35.000Z",
  "title": "Well-posedness and stability results for a quasilinear periodic Muskat problem",
  "authors": [
    "Anca-Voichita Matioc",
    "Bogdan-Vasile Matioc"
  ],
  "comment": "26 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the Muskat problem describing the spatially periodic motion of two fluids with equal viscosities under the effect of gravity in a vertical unbounded two-dimensional geometry. We first prove that the classical formulation of the problem is equivalent to a nonlocal and nonlinear evolution equation expressed in terms of singular integrals and having only the interface between the fluids as unknown. Secondly, we show that this evolution equation has a quasilinear structure, which is at a formal level not obvious, and we also disclose the parabolic character of the equation. Exploiting these aspects, we establish the local well-posedness of the problem for arbitrary initial data in $H^s(\\mathbb{S})$, with $s\\in(3/2,2)$, determine a new criterion for the global existence of solutions, and uncover a parabolic smoothing property. Besides, we prove that the zero steady-state solution is exponentially stable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-28T12:50:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B35",
      "35B65",
      "35K59",
      "35Q35",
      "42B20"
    ],
    "keywords": [
      "quasilinear periodic muskat problem",
      "stability results",
      "well-posedness",
      "nonlinear evolution equation",
      "arbitrary initial data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}