{
  "id": "2002.12605",
  "version": "v1",
  "published": "2020-02-28T09:17:56.000Z",
  "updated": "2020-02-28T09:17:56.000Z",
  "title": "On the global solvability of the axisymmetric Boussinesq system with critical regularity",
  "authors": [
    "Haroune Houamed",
    "Mohamed Zerguine"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "The current paper is principally motivated by establishing the global well-posedness to the three-dimensional Boussinesq system with zero diffusivity in the setting of axisymmetric flows without swirling with $v_0\\in H^{\\frac12}(\\mathbb{R}^3)\\cap \\dot{B}^{0}_{3,1}(\\mathbb{R}^3)$ and density $\\rho_0\\in L^2(\\mathbb{R}^3)\\cap \\dot{B}^{0}_{3,1}(\\mathbb{R}^3)$. This respectively enhances the two results recently accomplished in \\cite{Danchin-Paicu1, Hmidi-Rousset}. Our formalism is inspired, in particular for the first part from \\cite{Abidi} concerning the axisymmetric Navier-Stokes equations once $v_0\\in H^{\\frac12}(\\mathbb{R}^3)$ and external force $f\\in L^2_{loc}\\big(\\mathbb{R}_{+};H^{\\beta}(\\mathbb{R}^3)\\big)$, with $\\beta>\\frac14$. This latter regularity on $f$ which is the density in our context is helpless to achieve the global estimates for Boussinesq system. This technical defect forces us to deal once again with a similar proof to that of \\cite{Abidi} but with $f\\in L^{\\beta}_{loc}\\big(\\mathbb{R}_{+};L^2(\\mathbb{R}^3))$ for some $\\beta>4$. Second, we explore the gained regularity on the density by considering it as an external force in order to apply the study already obtained to the Boussinesq system.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-28T09:17:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "76D03",
      "76D05",
      "35B33",
      "35Q35"
    ],
    "keywords": [
      "axisymmetric boussinesq system",
      "global solvability",
      "critical regularity",
      "external force",
      "three-dimensional boussinesq system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}