{
  "id": "math/0612457",
  "version": "v1",
  "published": "2006-12-15T21:08:53.000Z",
  "updated": "2006-12-15T21:08:53.000Z",
  "title": "A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order",
  "authors": [
    "Michael Christ",
    "James Colliander",
    "Terence Tao"
  ],
  "comment": "22 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Solutions to the Cauchy problem for the one-dimensional cubic nonlinear Schr\\\"odinger equation on the real line are studied in Sobolev spaces $H^s$, for $s$ negative but close to 0. For smooth solutions there is an {\\em a priori} upper bound for the $H^s$ norm of the solution, in terms of the $H^s$ norm of the datum, for arbitrarily large data, for sufficiently short time. Weak solutions are constructed for arbitrary initial data in $H^s$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-12-15T21:08:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55"
    ],
    "keywords": [
      "nonlinear schrödinger equation",
      "sobolev spaces",
      "weak solutions",
      "priori bounds",
      "negative order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....12457C"
    }
  }
}