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

Weobtain improved regularity criteria for the axisymmetricweak 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.