{
  "id": "1003.0707",
  "version": "v4",
  "published": "2010-03-03T20:53:58.000Z",
  "updated": "2011-02-14T15:57:23.000Z",
  "title": "On stable self-similar blow up for equivariant wave maps",
  "authors": [
    "Roland Donninger"
  ],
  "comment": "Some minor changes according to the referee's suggestions, to appear in Comm. Pure Appl. Math",
  "journal": "Comm. Pure Appl. Math. 64 (2011), no. 8, 1029-1164",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider co--rotational wave maps from (3+1) Minkowski space into the three--sphere. This is an energy supercritical model which is known to exhibit finite time blow up via self-similar solutions. The ground state self--similar solution $f_0$ is known in closed form and based on numerics, it is supposed to describe the generic blow up behavior of the system. We prove that the blow up via $f_0$ is stable under the assumption that $f_0$ does not have unstable modes. This condition is equivalent to a spectral assumption for a linear second order ordinary differential operator. In other words, we reduce the problem of stable blow up to a linear ODE spectral problem. Although we are unable, at the moment, to verify the mode stability of $f_0$ rigorously, it is known that possible unstable eigenvalues are confined to a certain compact region in the complex plane. As a consequence, highly reliable numerical techniques can be applied and all available results strongly suggest the nonexistence of unstable modes, i.e., the assumed mode stability of $f_0$.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2011-02-14T15:57:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L05",
      "35B44",
      "35C06"
    ],
    "keywords": [
      "equivariant wave maps",
      "stable self-similar blow",
      "second order ordinary differential operator",
      "linear second order ordinary differential",
      "ground state self-similar solution"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1002/cpa.20366"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 847864,
      "adsabs": "2010arXiv1003.0707D"
    }
  }
}