{
  "id": "2008.01272",
  "version": "v1",
  "published": "2020-08-04T02:02:03.000Z",
  "updated": "2020-08-04T02:02:03.000Z",
  "title": "Regularity for a special case of two-phase Hele-Shaw flow via parabolic integro-differential equations",
  "authors": [
    "Farhan Abedin",
    "Russell W. Schwab"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We establish that the $C^{1,\\gamma}$ regularity theory for translation invariant fractional order parabolic integro-differential equations (via Krylov-Safonov estimates) gives an improvement of regularity mechanism for solutions to a special case of a two-phase free boundary flow related to Hele-Shaw. The special case is due to both a graph assumption on the free boundary of the flow and an assumption that the free boundary is $C^{1,\\text{Dini}}$ in space. The free boundary then must immediately become $C^{1,\\gamma}$ for a universal $\\gamma$ depending upon the Dini modulus of the gradient of the graph. These results also apply to one-phase problems of the same type.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-04T02:02:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "two-phase hele-shaw flow",
      "special case",
      "fractional order parabolic integro-differential",
      "order parabolic integro-differential equations",
      "invariant fractional order parabolic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}