{
  "id": "1407.1479",
  "version": "v4",
  "published": "2014-07-06T10:30:21.000Z",
  "updated": "2014-08-20T16:08:02.000Z",
  "title": "On the impossibility of finite-time splash singularities for vortex sheets",
  "authors": [
    "Daniel Coutand",
    "Steve Shkoller"
  ],
  "comment": "29 pages, 8 figures, added Remarks 1,2,4,5 in Sections 4 and 5",
  "categories": [
    "math.AP"
  ],
  "abstract": "In fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. By means of elementary arguments, we prove that such a singularity cannot occur in finite time for vortex sheet evolution, i.e. for the two-phase incompressible Euler equations. We prove this by contradiction; we assume that a splash singularity does indeed occur in finite time. Based on this assumption, we find precise blow-up rates for the components of the velocity gradient which, in turn, allow us to characterize the geometry of the evolving interface just prior to self-intersection. The constraints on the geometry then lead to an impossible outcome, showing that our assumption of a finite-time splash singularity was false.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-08-01T18:42:35.000Z",
      "comment": "27 pages, 8 figures, typos corrects, simplified the presentation and added a figure in Section 7.3",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2014-08-20T16:08:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q35"
    ],
    "keywords": [
      "finite-time splash singularity",
      "finite time",
      "impossibility",
      "interface splash singularity occurs",
      "vortex sheet evolution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.1479C"
    }
  }
}