{
  "id": "2405.16670",
  "version": "v1",
  "published": "2024-05-26T19:21:26.000Z",
  "updated": "2024-05-26T19:21:26.000Z",
  "title": "On the regularity of axially-symmetric solutions to the incompressible Navier-Stokes equations in a cylinder",
  "authors": [
    "W. S. Ożański",
    "W. Zajączkowski"
  ],
  "comment": "30 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the axisymmetric Navier-Stokes equations in a finite cylinder $\\Omega\\subset\\mathbb{R}^3$. We assume that $v_r$, $v_\\varphi$, $\\omega_\\varphi$ vanish on the lateral boundary $\\partial \\Omega$ of the cylinder, and that $v_z$, $\\omega_\\varphi$, $\\partial_z v_\\varphi$ vanish on the top and bottom parts of the boundary $\\partial \\Omega$, where we used standard cylindrical coordinates, and we denoted by $\\omega =\\mathrm{curl}\\, v$ the vorticity field. We use estimates and $H^3$ Sobolev estimate on the modified stream function to derive three order-reduction estimates. These enable one to reduce the order of the nonlinear estimates of the equations, and help observe that the solutions to the equations is ``almost regular''. We use the order-reduction estimates to show that the solution to the equations remains regular as long as, for any $p\\in (6,\\infty)$, $\\| v_\\varphi \\|_{L^\\infty_t L^p_x}/\\| v_\\varphi \\|_{L^\\infty_t L^\\infty_x}$ remains bounded below by a positive number.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-26T19:21:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "incompressible navier-stokes equations",
      "axially-symmetric solutions",
      "order-reduction estimates",
      "regularity",
      "equations remains regular"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}