{
  "id": "math/0609591",
  "version": "v1",
  "published": "2006-09-21T05:55:03.000Z",
  "updated": "2006-09-21T05:55:03.000Z",
  "title": "Asymptotic stability of harmonic maps under the Schrödinger flow",
  "authors": [
    "Stephen Gustafson",
    "Kyungkeun Kang",
    "Tai-Peng Tsai"
  ],
  "comment": "41 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "For Schr\\\"odinger maps from $\\R^2\\times\\R^+$ to the 2-sphere $\\S^2$, it is not known if finite energy solutions can form singularities (``blowup'') in finite time. We consider equivariant solutions with energy near the energy of the two-parameter family of equivariant harmonic maps. We prove that if the topological degree of the map is at least four, blowup does {\\it not} occur, and global solutions converge (in a dispersive sense -- i.e. scatter) to a fixed harmonic map as time tends to infinity. The proof uses, among other things, a time-dependent splitting of the solution, the ``generalized Hasimoto transform\", and Strichartz (dispersive) estimates for a certain two space-dimensional linear Schr\\\"odinger equation whose potential has critical power spatial singularity and decay. Along the way, we establish an energy-space local well-posedness result for which the existence time is determined by the length-scale of a nearby harmonic map.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-09-21T05:55:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55",
      "35B40"
    ],
    "keywords": [
      "asymptotic stability",
      "schrödinger flow",
      "energy-space local well-posedness result",
      "critical power spatial singularity",
      "nearby harmonic map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......9591G"
    }
  }
}