{
  "id": "1011.5066",
  "version": "v2",
  "published": "2010-11-23T11:38:15.000Z",
  "updated": "2010-11-27T11:53:07.000Z",
  "title": "A Liouville Theorem for the Axially-symmetric Navier-Stokes Equations",
  "authors": [
    "Zhen Lei",
    "Qi S. Zhang"
  ],
  "comment": "1. We give a partial proof of a conjecture on Liouville property by Koch-Nadirashvili-Seregin-Sverak in \\cite{KNSS} and Seregin-Sverak in \\cite{SS}. We also solved an open question raised by Koch and Tataru in \\cite{KochTataru} in the axi-symmetric case. 2. Comparing with the previous version, one reference is added",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $v(x, t)= v^r e_r + v^\\theta e_\\theta + v^z e_z$ be a solution to the three-dimensional incompressible axially-symmetric Navier-Stokes equations. Denote by $b = v^r e_r + v^z e_z$ the radial-axial vector field. Under a general scaling invariant condition on $b$, we prove that the quantity $\\Gamma = r v^\\theta$ is H\\\"older continuous at $r = 0$, $t = 0$. As an application, we give a partial proof of a conjecture on Liouville property by Koch-Nadirashvili-Seregin-Sverak in \\cite{KNSS} and Seregin-Sverak in \\cite{SS}. As another application, we prove that if $b \\in L^\\infty([0, T], BMO^{-1})$, then $v$ is regular. This provides an answer to an open question raised by Koch and Tataru in \\cite{KochTataru} about the uniqueness and regularity of Navier-Stokes equations in the axially-symmetric case.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-11-27T11:53:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "liouville theorem",
      "three-dimensional incompressible axially-symmetric navier-stokes equations",
      "general scaling invariant condition",
      "radial-axial vector field",
      "application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1011.5066L"
    }
  }
}