{
  "id": "1708.01568",
  "version": "v1",
  "published": "2017-08-04T16:05:24.000Z",
  "updated": "2017-08-04T16:05:24.000Z",
  "title": "On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities",
  "authors": [
    "Tadahiro Oh",
    "Mamoru Okamoto",
    "Oana Pocovnicu"
  ],
  "comment": "44 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the Cauchy problem for the nonlinear Schr\\\"odinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on $\\mathbb{R}^d$, $d=5,6$, and energy-critical NLS without gauge invariance and prove that they are almost surely locally well-posed with respect to randomized initial data below the energy space. We also study the long time behavior of solutions to these equations: (i) we prove almost sure global well-posedness of the (standard) energy-critical NLS on $\\mathbb{R}^d$, $d = 5, 6$, in the defocusing case, and (ii) we present a probabilistic construction of finite time blowup solutions to the energy-critical NLS without gauge invariance below the energy space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-04T16:05:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55"
    ],
    "keywords": [
      "nonlinear schrödinger equations",
      "non-algebraic nonlinearities",
      "probabilistic well-posedness",
      "energy-critical nls",
      "finite time blowup solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}