{
  "id": "2203.13567",
  "version": "v1",
  "published": "2022-03-25T10:52:14.000Z",
  "updated": "2022-03-25T10:52:14.000Z",
  "title": "A new reformulation of the Muskat problem with surface tension",
  "authors": [
    "Anca--Voichita Matioc",
    "Bogdan--Vasile Matioc"
  ],
  "comment": "23 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Two formulas that connect the derivatives of the double layer potential and of a related singular integral operator evaluated at some density $\\vartheta$ to the $L_2$-adjoints of these operators evaluated at the density $\\vartheta'$ are used to recast the Muskat problem with surface tension and general viscosities as a system of equations with nonlinearities expressed in terms of the $L_2$-adjoints of these operators. An advantage of this formulation is that the nonlinearities appear now as a derivative. This aspect and abstract quasilinear parabolic theory are then exploited to establish a local well-posedness result in all subcritical Sobolev spaces $W^s_p(\\mathbb{R})$ with $p\\in(1,\\infty)$ and $s\\in (1+1/p,2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-25T10:52:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R37",
      "35K59",
      "35K93",
      "35Q35",
      "42B20"
    ],
    "keywords": [
      "surface tension",
      "muskat problem",
      "abstract quasilinear parabolic theory",
      "reformulation",
      "related singular integral operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}