{
  "id": "1203.6374",
  "version": "v1",
  "published": "2012-03-28T20:42:53.000Z",
  "updated": "2012-03-28T20:42:53.000Z",
  "title": "Sharp local well-posedness for the \"good\" Boussinesq equation",
  "authors": [
    "Nobu Kishimoto"
  ],
  "comment": "40 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In the present article, we prove the sharp local well-posedness and ill-posedness results for the \"good\" Boussinesq equation on $\\mathbb{T}$; the initial value problem is locally well-posed in $H^{-1/2}(\\mathbb{T})$ and ill-posed in $H^s(\\mathbb{T})$ for $s<-1/2$. Well-posedness result is obtained from reduction of the problem into a quadratic nonlinear Schr\\\"odinger equation and the contraction argument in suitably modified $X^{s,b}$ spaces. The proof of the crucial bilinear estimates in these spaces, especially in the lowest regularity, rely on some bilinear estimates for one dimensional periodic functions in $X^{s,b}$ spaces, which are generalization of the bilinear refinement of the $L^4$ Strichartz estimate on $\\mathbb{R}$. Our result improves the known local well-posedness in $H^s(\\mathbb{T})$ with $s>-3/8$ given by Oh and Stefanov (2012) to the regularity threshold $H^{-1/2}(\\mathbb{T})$. Similar ideas also establish the sharp local well-posedness in $H^{-1/2}(\\mathbb{R})$ and ill-posedness below $H^{-1/2}$ for the nonperiodic case, which improves the result of Tsugawa and the author (2010) in $H^s(\\mathbb{R})$ with $s>-1/2$ to the limiting regularity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-03-28T20:42:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55"
    ],
    "keywords": [
      "sharp local well-posedness",
      "boussinesq equation",
      "crucial bilinear estimates",
      "initial value problem",
      "dimensional periodic functions"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1016/j.jde.2012.12.008",
      "journal": "Journal of Differential Equations",
      "year": 2013,
      "volume": 254,
      "number": 6,
      "pages": 2393
    },
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013JDE...254.2393K"
    }
  }
}