{
  "id": "2201.09343",
  "version": "v1",
  "published": "2022-01-23T18:45:39.000Z",
  "updated": "2022-01-23T18:45:39.000Z",
  "title": "Sharp Interface Limit for a Navier-Stokes/Allen-Cahn System with Different Viscosities",
  "authors": [
    "Helmut Abels",
    "Mingwen Fei"
  ],
  "comment": "44 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We discuss the sharp interface limit of a coupled Navier-Stokes/Allen-Cahn system in a two dimensional, bounded and smooth domain, when a parameter $\\varepsilon>0$ that is proportional to the thickness of the diffuse interface tends to zero rigorously. We prove convergence of the solutions of the Navier-Stokes/Allen-Cahn system to solutions of a sharp interface model, where the interface evolution is given by the mean curvature flow with an additional convection term coupled to a two-phase Navier-Stokes system with surface tension. This is done by constructing an approximate solution from the limiting system via matched asymptotic expansions together with a novel Ansatz for the highest order term, and then estimating its difference with the real solution with the aid of a refined spectral estimate of the linearized Allen-Cahn operator near the approximate solution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-23T18:45:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "76T99",
      "35Q30",
      "35Q35",
      "35R35",
      "76D05",
      "76D45"
    ],
    "keywords": [
      "sharp interface limit",
      "navier-stokes/allen-cahn system",
      "viscosities",
      "approximate solution",
      "diffuse interface tends"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}