{
  "id": "1402.1591",
  "version": "v1",
  "published": "2014-02-07T10:22:35.000Z",
  "updated": "2014-02-07T10:22:35.000Z",
  "title": "Recent progress in smoothing estimates for evolution equations",
  "authors": [
    "Michael Ruzhansky",
    "Mitsuru Sugimoto"
  ],
  "comment": "13 pages",
  "journal": "in Progress in Partial Differential Equations, pp. 287-302, Springer Proceedings in Mathematics & Statistics, Vol. 44, 2013",
  "doi": "10.1007/978-3-319-00125-8_13",
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper is a survey article of results and arguments from several of authors' papers, and it describes a new approach to global smoothing problems for dispersive and non-dispersive evolution equations based on ideas of comparison principle and canonical transforms. For operators $a(D_x)$ of order $m$ satisfying the dispersiveness condition $\\nabla a(\\xi)\\neq0$, a range of smoothing estimates is established. Especially, time-global smoothing estimates for the operator $a(D_x)$ with lower order terms are the benefit of our new method. These estimates are known to fail for general non-dispersive operators. For the case when the dispersiveness breaks, we suggest a modification of the smoothing estimate. It is equivalent to the usual estimate in the dispersive case and is also invariant under canonical transformations for the operator $a(D_x)$. Moreover, it does continue to hold for a variety of non-dispersive operators $a(D_x)$, where $\\nabla a(\\xi)$ may become zero on some set. It is interesting that this method allows us to carry out a global microlocal reduction of equations to the translation invariance property of the Lebesgue measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-07T10:22:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lower order terms",
      "global microlocal reduction",
      "translation invariance property",
      "dispersiveness condition",
      "time-global smoothing estimates"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.1591R"
    }
  }
}