{
  "id": "math/0605023",
  "version": "v3",
  "published": "2006-04-30T15:07:14.000Z",
  "updated": "2008-08-22T20:53:07.000Z",
  "title": "On the Formation of Singularities in the Critical O(3) Sigma-Model",
  "authors": [
    "Igor Rodnianski",
    "Jacob Sterbenz"
  ],
  "comment": "51 pages added remarks and references corrected computation of the constant in the Appendix A, leading to the stable blow up with the rate bounded from above by a log-modified self-similar asymptotic and from below by a self-similar rate",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the phenomena of energy concentration for the critical O(3) sigma model, also known as the wave map flow from R^{2+1} Minkowski space into the sphere S^2. We establish rigorously and constructively existence of a set of smooth initial data resulting in a dynamic finite time formation of singularities. The construction and analysis is done in the context of the k-equivariant symmetry reduction, and we restrict to maps with homotopy class k>3. The concentration mechanism we uncover is essentially due to a resonant self-focusing (shrinking) of a corresponding harmonic map. We show that the phenomenon is generic (e.g. in certain Sobolev spaces) and persists under small perturbations of initial data, while the resulting blowup is bounded by a log-modified self-similar asymptotic.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2008-08-22T20:53:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "singularities",
      "sigma-model",
      "dynamic finite time formation",
      "k-equivariant symmetry reduction",
      "wave map flow"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 715846,
      "adsabs": "2006math......5023R"
    }
  }
}