{
  "id": "2005.14128",
  "version": "v1",
  "published": "2020-05-28T16:24:39.000Z",
  "updated": "2020-05-28T16:24:39.000Z",
  "title": "Non-Uniqueness of Bubbling for Wave Maps",
  "authors": [
    "Max Engelstein",
    "Dana Mendelson"
  ],
  "comment": "26 pages, two figures. Comments welcome",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "We consider wave maps from $\\mathbb R^{2+1}$ to a $C^\\infty$-smooth Riemannian manifold, $\\mathcal N$. Such maps can exhibit energy concentration, and at points of concentration, it is known that the map (suitably rescaled and translated) converges weakly to a harmonic map, known as a bubble. We give an example of a wave map which exhibits a type of non-uniqueness of bubbling. In particular, we exhibit a continuum of different bubbles at the origin, each of which arise as the weak limit along a different sequence of times approaching the blow-up time. This is the first known example of non-uniqueness of bubbling for dispersive equations. Our construction is inspired by the work of Peter Topping [Topping 2004], who demonstrated a similar phenomena can occur in the setting of harmonic map heat flow, and our mechanism of non-uniqueness is the same 'winding' behavior exhibited in that work.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-28T16:24:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "wave map",
      "non-uniqueness",
      "harmonic map heat flow",
      "smooth riemannian manifold",
      "energy concentration"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}