{
  "id": "1502.00937",
  "version": "v1",
  "published": "2015-02-03T17:33:26.000Z",
  "updated": "2015-02-03T17:33:26.000Z",
  "title": "Scattering in the energy space for the NLS with variable coefficients",
  "authors": [
    "Biagio Cassano",
    "Piero D'Ancona"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the NLS with variable coefficients in dimension $n\\ge3$ \\begin{equation*} i \\partial_t u - Lu +f(u)=0, \\qquad Lv=\\nabla^{b}\\cdot(a(x)\\nabla^{b}v)-c(x)v, \\qquad \\nabla^{b}=\\nabla+ib(x), \\end{equation*} on $\\mathbb{R}^{n}$ or more generally on an exterior domain with Dirichlet boundary conditions, for a gauge invariant, defocusing nonlinearity of power type $f(u)\\simeq|u|^{\\gamma-1}u$. We assume that $L$ is a small, long range perturbation of $\\Delta$, plus a potential with a large positive part. The first main result of the paper is a bilinear smoothing (interaction Morawetz) estimate for the solution. As an application, under the conditional assumption that Strichartz estimates are valid for the linear flow $e^{itL}$, we prove global well posedness in the energy space for subcritical powers $\\gamma<1+\\frac{4}{n-2}$, and scattering provided $\\gamma>1+\\frac4n$. When the domain is $\\mathbb{R}^{n}$, by extending the Strichartz estimates due to Tataru [Tataru08], we prove that the conditional assumption is satisfied and deduce well posedness and scattering in the energy space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-03T17:33:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L70",
      "58J45"
    ],
    "keywords": [
      "energy space",
      "variable coefficients",
      "scattering",
      "strichartz estimates",
      "conditional assumption"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150200937C"
    }
  }
}