{
  "id": "math/0701497",
  "version": "v2",
  "published": "2007-01-18T01:06:54.000Z",
  "updated": "2007-01-30T05:30:47.000Z",
  "title": "Cauchy problem of nonlinear Schrödinger equation with Cauchy problem of nonlinear Schrödinger equation with initial data in Sobolev space $W^{s,p}$ for $p<2$",
  "authors": [
    "Yi Zhou"
  ],
  "comment": "12 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we consider in $R^n$ the Cauchy problem for nonlinear Schr\\\"odinger equation with initial data in Sobolev space $W^{s,p}$ for $p<2$. It is well known that this problem is ill posed. However, We show that after a linear transformation by the linear semigroup the problem becomes locally well posed in $W^{s,p}$ for $\\frac{2n}{n+1}<p<2$ and $s>n(1-\\frac{1}{p})$. Moreover, we show that in one space dimension, the problem is locally well posed in $L^p$ for any $1<p<2$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-01-30T05:30:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q35"
    ],
    "keywords": [
      "nonlinear schrödinger equation",
      "cauchy problem",
      "sobolev space",
      "initial data",
      "linear transformation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......1497Z"
    }
  }
}