{
  "id": "1304.0996",
  "version": "v2",
  "published": "2013-04-03T15:59:19.000Z",
  "updated": "2014-03-17T16:38:18.000Z",
  "title": "The initial value problem for the binormal flow with rough data",
  "authors": [
    "Valeria Banica",
    "Luis Vega"
  ],
  "comment": "34 pages, 3 figures, revised version, to appear in Ann. Sci. \\'Ec. Norm. Sup\\'er. (4)",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this article we consider the initial value problem of the binormal flow with initial data given by curves that are regular except at one point where they have a corner. We prove that under suitable conditions on the initial data a unique regular solution exists for strictly positive and strictly negative times. Moreover, this solution satisfies a weak version of the equation for all times and can be seen as a perturbation of a suitably chosen self-similar solution. Conversely, we also prove that if at time t = 1 a small regular perturbation of a self-similar solution is taken as initial condition then there exists a unique solution that at time t = 0 is regular except at a point where it has a corner with the same angle as the one of the self-similar solution. This solution can be extended for negative times. The proof uses the full strength of the previous papers [9], [2], [3] and [4] on the study of small perturbations of self-similar solutions. A compactness argument is used to avoid the weighted conditions we needed in [4], as well as a more refined analysis of the asymptotic in time and in space of the tangent and normal vectors.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-03-17T16:38:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "initial value problem",
      "binormal flow",
      "rough data",
      "initial data",
      "suitably chosen self-similar solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.0996B"
    }
  }
}