On the regularity of the axisymmetric solutions of the Navier-Stokes equations

We obtain improved regularity criteria for the axisymmetric weak solutions of the three dimensional Navier-Stokes equations with nonzero swirl. In particular we prove that the integrability of single component of vorticity or velocity fields, in terms of norms with zero scaling dimension give sufficient conditions for the regularity of weak solutions. To obtain these criteria we derive new a priori estimates for the axisymmetric smooth solutions of the Navier-Stokes equations. Received: 11 April 2000; in final firm: 26 November 2000 / Published online: 28 February 2002