{
  "id": "1611.04422",
  "version": "v1",
  "published": "2016-11-14T15:43:45.000Z",
  "updated": "2016-11-14T15:43:45.000Z",
  "title": "Sharp Interface Limit for a Stokes/Allen-Cahn System",
  "authors": [
    "Helmut Abels",
    "YuNing Liu"
  ],
  "comment": "62 pages",
  "categories": [
    "math.AP",
    "physics.flu-dyn"
  ],
  "abstract": "We consider the sharp interface limit of a coupled Stokes/Allen-Cahn system, when a parameter $\\varepsilon>0$ that is proportional to the thickness of the diffuse interface tends to zero, in a two dimensional bounded domain. For sufficiently small times we prove convergence of the solutions of the Stokes/Allen-Cahn system to solutions of a sharp interface model, where the interface evolution is given by the mean curvature equation with an additional convection term coupled to a two-phase Stokes system with an additional contribution to the stress tensor, which describes the capillary stress. To this end we construct a suitable approximation of the solution of the Stokes/Allen-Cahn system, using three levels of the terms in the formally matched asymptotic calculations, and estimate the difference with the aid of a suitable refinement of a spectral estimate due to Chen for the linearized Allen-Cahn operator. Moreover, a careful treatment of the coupling terms is needed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-14T15:43:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "76T99",
      "35Q30",
      "35Q35",
      "35R35",
      "76D05",
      "76D45"
    ],
    "keywords": [
      "sharp interface limit",
      "stokes/allen-cahn system",
      "two-phase stokes system",
      "additional convection term",
      "mean curvature equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 62,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}