{
  "id": "0805.3378",
  "version": "v1",
  "published": "2008-05-22T00:58:22.000Z",
  "updated": "2008-05-22T00:58:22.000Z",
  "title": "Global well-posedness and scattering for the defocusing $H^{\\frac12}$-subcritical Hartree equation in $\\mathbb{R}^d$",
  "authors": [
    "Changxing Miao",
    "Guixiang Xu",
    "Lifeng Zhao"
  ],
  "comment": "24 pages,1 figure",
  "journal": "Ann.I. H. Poincare-AN 26 (2009) 1831-1852",
  "doi": "10.1016/j.anihpc.2009.01.003",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove the global well-posedness and scattering for the defocusing $H^{\\frac12}$-subcritical (that is, $2<\\gamma<3$) Hartree equation with low regularity data in $\\mathbb{R}^d$, $d\\geq 3$. Precisely, we show that a unique and global solution exists for initial data in the Sobolev space $H^s\\big(\\mathbb{R}^d\\big)$ with $s>4(\\gamma-2)/(3\\gamma-4)$, which also scatters in both time directions. This improves the result in \\cite{ChHKY}, where the global well-posedness was established for any $s>\\max\\big(1/2,4(\\gamma-2)/(3\\gamma-4)\\big)$. The new ingredients in our proof are that we make use of an interaction Morawetz estimate for the smoothed out solution $Iu$, instead of an interaction Morawetz estimate for the solution $u$, and that we make careful analysis of the monotonicity property of the multiplier $m(\\xi)\\cdot < \\xi>^p$. As a byproduct of our proof, we obtain that the $H^s$ norm of the solution obeys the uniform-in-time bounds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-05-22T00:58:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q40",
      "35Q55",
      "47J35"
    ],
    "keywords": [
      "global well-posedness",
      "subcritical hartree equation",
      "interaction morawetz estimate",
      "low regularity data",
      "defocusing"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009AnIHP..26.1831M"
    }
  }
}