{
  "id": "1212.4222",
  "version": "v2",
  "published": "2012-12-18T03:45:41.000Z",
  "updated": "2013-01-29T12:38:29.000Z",
  "title": "Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity",
  "authors": [
    "Changxing Miao",
    "Xiaoxin Zheng"
  ],
  "comment": "32pages. arXiv admin note: text overlap with arXiv:0908.0894 by other authors",
  "journal": "J. Math. Pures Appl. 101 (2014) 842-872",
  "doi": "10.1016/j.matpur.2013.10.007",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In this paper, we are concerned with the tridimensional anisotropic Boussinesq equations which can be described by {equation*} {{array}{ll} (\\partial_{t}+u\\cdot\\nabla)u-\\kappa\\Delta_{h} u+\\nabla \\Pi=\\rho e_{3},\\quad(t,x)\\in\\mathbb{R}^{+}\\times\\mathbb{R}^{3}, (\\partial_{t}+u\\cdot\\nabla)\\rho=0, \\text{div}u=0. {array}. {equation*} Under the assumption that the support of the axisymmetric initial data $\\rho_{0}(r,z)$ does not intersect the axis $(Oz)$, we prove the global well-posedness for this system with axisymmetric initial data. We first show the growth of the quantity $\\frac\\rho r$ for large time by taking advantage of characteristic of transport equation. This growing property together with the horizontal smoothing effect enables us to establish $H^1$-estimate of the velocity via the $L^2$-energy estimate of velocity and the Maximum principle of density. Based on this, we further establish the estimate for the quantity $\\|\\omega(t)\\|_{\\sqrt{\\mathbb{L}}}:=\\sup_{2\\leq p<\\infty}\\frac{\\norm{\\omega(t)}_{L^p(\\mathbb{R}^3)}}{\\sqrt{p}}<\\infty$ which implies $\\|\\nabla u(t)\\|_{\\mathbb{L}^{3/2}}:=\\sup_{2\\leq p<\\infty}\\frac{\\norm{\\nabla u(t)}_{L^p(\\mathbb{R}^3)}}{p\\sqrt{p}}<\\infty$. However, this regularity for the flow admits forbidden singularity since $ \\mathbb{L}$ (see \\eqref{eq-kl} for the definition) seems be the minimum space for the gradient vector field $u(x,t)$ ensuring uniqueness of flow. To bridge this gap, we exploit the space-time estimate about $ \\sup_{2\\leq p<\\infty}\\int_0^t\\frac{\\|\\nabla u(\\tau)\\|_{L^p(\\mathbb{R}^3)}}{\\sqrt{p}}\\mathrm{d}\\tau<\\infty$ by making good use of the horizontal smoothing effect and micro-local techniques. The global well-posedness for the large initial data is achieved by establishing a new type space-time logarithmic inequality.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-01-29T12:38:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B33",
      "35Q35",
      "76D03",
      "76D05"
    ],
    "keywords": [
      "axisymmetric boussinesq system",
      "global well-posedness",
      "horizontal viscosity",
      "axisymmetric initial data",
      "horizontal smoothing effect"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.4222M"
    }
  }
}