{
  "id": "0807.2676",
  "version": "v4",
  "published": "2008-07-16T23:53:15.000Z",
  "updated": "2009-02-23T23:03:45.000Z",
  "title": "Global existence and uniqueness results for weak solutions of the focusing mass-critical non-linear Schrödinger equation",
  "authors": [
    "Terence Tao"
  ],
  "comment": "25 pages, 1 table, to appear, Analysis & PDE. Typo in definition of semi-Strichartz solution corrected",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the focusing mass-critical NLS $iu_t + \\Delta u = - |u|^{4/d} u$ in high dimensions $d \\geq 4$, with initial data $u(0) = u_0$ having finite mass $M(u_0) = \\int_{\\R^d} |u_0(x)|^2 dx < \\infty$. It is well known that this problem admits unique (but not global) strong solutions in the Strichartz class $C^0_{t,\\loc} L^2_x \\cap L^2_{t,\\loc} L^{2d/(d-2)}_x$, and also admits global (but not unique) weak solutions in $L^\\infty_t L^2_x$. In this paper we introduce an intermediate class of solution, which we call a \\emph{semi-Strichartz class solution}, for which one does have global existence and uniqueness in dimensions $d \\geq 4$. In dimensions $d \\geq 5$ and assuming spherical symmetry, we also show the equivalence of the Strichartz class and the strong solution class (and also of the semi-Strichartz class and the semi-strong solution class), thus establishing ``unconditional'' uniqueness results in the strong and semi-strong classes. With these assumptions we also characterise these solutions in terms of the continuity properties of the mass function $t \\mapsto M(u(t))$.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2009-02-23T23:03:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q30"
    ],
    "keywords": [
      "focusing mass-critical non-linear schrödinger equation",
      "uniqueness results",
      "weak solutions",
      "global existence",
      "strong solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0807.2676T"
    }
  }
}