{
  "id": "2003.05011",
  "version": "v1",
  "published": "2020-03-10T21:39:07.000Z",
  "updated": "2020-03-10T21:39:07.000Z",
  "title": "Sharp well-posedness for the cubic NLS and mKdV in $H^s(\\mathbb R)$",
  "authors": [
    "Benjamin Harrop-Griffiths",
    "Rowan Killip",
    "Monica Visan"
  ],
  "comment": "81 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove that the cubic nonlinear Schr\\\"odinger equation (both focusing and defocusing) is globally well-posed in $H^s(\\mathbb R)$ for any regularity $s>-\\frac12$. Well-posedness has long been known for $s\\geq 0$, see [51], but not previously for any $s<0$. The scaling-critical value $s=-\\frac12$ is necessarily excluded here, since instantaneous norm inflation is known to occur [11, 38, 46]. We also prove (in a parallel fashion) well-posedness of the real- and complex-valued modified Korteweg-de Vries equations in $H^s(\\mathbb R)$ for any $s>-\\frac12$. The best regularity achieved previously was $s\\geq \\tfrac14$; see [15, 24, 32, 38]. An essential ingredient in our arguments is the demonstration of a local smoothing effect for both equations, with a gain of derivatives matching that of the underlying linear equation. This in turn rests on the discovery of a one-parameter family of microscopic conservation laws that remain meaningful at this low regularity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-10T21:39:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55",
      "35Q53"
    ],
    "keywords": [
      "cubic nls",
      "sharp well-posedness",
      "complex-valued modified korteweg-de vries equations",
      "microscopic conservation laws",
      "low regularity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 81,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}