{
  "id": "math/0602560",
  "version": "v1",
  "published": "2006-02-24T20:49:32.000Z",
  "updated": "2006-02-24T20:49:32.000Z",
  "title": "Global Well-Posedness for a periodic nonlinear Schrödinger equation in 1D and 2D",
  "authors": [
    "Daniela De Silva",
    "Nataša Pavlović",
    "Gigliola Staffilani",
    "Nikolaos Tzirakis"
  ],
  "comment": "28 pages, 3 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "The initial value problem for the $L^{2}$ critical semilinear Schr\\\"odinger equation with periodic boundary data is considered. We show that the problem is globally well posed in $H^{s}({\\Bbb T^{d}})$, for $s>4/9$ and $s>2/3$ in 1D and 2D respectively, confirming in 2D a statement of Bourgain in \\cite{bo2}. We use the ``$I$-method''. This method allows one to introduce a modification of the energy functional that is well defined for initial data below the $H^{1}({\\Bbb T^{d}})$ threshold. The main ingredient in the proof is a \"refinement\" of the Strichartz's estimates that hold true for solutions defined on the rescaled space, $\\Bbb T^{d}_{\\lambda} = \\Bbb R^{d}/{\\lambda \\Bbb Z^{d}}$, $d=1,2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-02-24T20:49:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55",
      "35A05"
    ],
    "keywords": [
      "periodic nonlinear schrödinger equation",
      "global well-posedness",
      "initial value problem",
      "periodic boundary data",
      "hold true"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......2560D"
    }
  }
}