{
  "id": "1502.06180",
  "version": "v1",
  "published": "2015-02-22T05:32:24.000Z",
  "updated": "2015-02-22T05:32:24.000Z",
  "title": "Global well-posedness of the 2D Boussinesq equations with vertical dissipation",
  "authors": [
    "Jinkai Li",
    "Edriss S. Titi"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove the global well-posedness of the two-dimensional Boussinesq equations with only vertical dissipation. The initial data $(u_0,\\theta_0)$ are required to be only in the space $X=\\{f\\in L^2(\\mathbb R^2)\\,|\\,\\partial_xf\\in L^2(\\mathbb R^2)\\}$, and thus our result generalizes that in [C. Cao, J. Wu, Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation, Arch. Rational Mech. Anal., Vol. 208 (2013), 985-1004], where the initial data are assumed to be in $H^2(\\mathbb R^2)$. The assumption on the initial data is at the minimal level that is required to guarantee the uniqueness of the solutions. A logarithmic type limiting Sobolev embedding inequality for the $L^\\infty(\\mathbb R^2)$ norm, in terms of anisotropic Sobolev norms, and a logarithmic type Gronwall inequality are established to obtain the global in time a priori estimates, which guarantee the local solution to be a global one.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-22T05:32:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A01",
      "35B45",
      "35Q86",
      "76D03",
      "76D09"
    ],
    "keywords": [
      "2d boussinesq equations",
      "vertical dissipation",
      "global well-posedness",
      "limiting sobolev embedding inequality",
      "initial data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}