{
  "id": "1310.7166",
  "version": "v2",
  "published": "2013-10-27T07:05:58.000Z",
  "updated": "2014-04-17T02:44:19.000Z",
  "title": "Global well-posedness for the nonlinear Schrödinger equation with derivative in energy space",
  "authors": [
    "Yifei Wu"
  ],
  "comment": "To appear in Analysis & PDE. We add some references, and change some expressions in English",
  "journal": "Analysis & PDE, 6 (8), 1989--2002 (2013)",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we prove that there exists some small $\\varepsilon_*>0$, such that the derivative nonlinear Schr\\\"{o}dinger equation (DNLS) is global well-posedness in the energy space, provided that the initial data $u_0\\in H^1(\\mathbb{R})$ satisfies $\\|u_0\\|_{L^2}<\\sqrt{2\\pi}+\\varepsilon_*$. This result shows us that there are no blow up solutions whose masses slightly exceed $2\\pi$, even if their energies are negative. This phenomenon is much different from the behavior of nonlinear Schr\\\"odinger equation with critical nonlinearity. The technique is a variational argument together with the momentum conservation law. Further, for the DNLS on half-line $\\mathbb{R}^+$, we show the blow-up for the solution with negative energy.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-04-17T02:44:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55",
      "35A01",
      "35B44"
    ],
    "keywords": [
      "nonlinear schrödinger equation",
      "energy space",
      "global well-posedness",
      "derivative",
      "momentum conservation law"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.7166W"
    }
  }
}