{
  "id": "2101.12274",
  "version": "v1",
  "published": "2021-01-28T20:58:15.000Z",
  "updated": "2021-01-28T20:58:15.000Z",
  "title": "On the well-posedness problem for the derivative nonlinear Schrödinger equation",
  "authors": [
    "Rowan Killip",
    "Maria Ntekoume",
    "Monica Visan"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the derivative nonlinear Schr\\\"odinger equation in one space dimension, posed both on the line and on the circle. This model is known to be completely integrable and $L^2$-critical with respect to scaling. The first question we discuss is whether ensembles of orbits with $L^2$-equicontinuous initial data remain equicontinuous under evolution. We prove that this is true under the restriction $M(q)=\\int |q|^2 < 4\\pi$. We conjecture that this restriction is unnecessary. Further, we prove that the problem is globally well-posed for initial data in $H^{1/6}$ under the same restriction on $M$. Moreover, we show that this restriction would be removed by a successful resolution of our equicontinuity conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-28T20:58:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55"
    ],
    "keywords": [
      "derivative nonlinear schrödinger equation",
      "well-posedness problem",
      "initial data remain equicontinuous",
      "restriction",
      "equicontinuous initial data remain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}