On some regularity criteria for axisymmetric Navier-Stokes equations

We point out some criteria that imply regularity of axisymmetric solutions to Navier-Stokes equations. We show that boundedness of $\mathrm{\parallel }{v}_r\mathrm{/}\sqrt{r^3}{\mathrm{\parallel }}_{L_2({\mathrm{R}}^3\times (0,T))}$ as well as boundedness of $\mathrm{\parallel }{\omega }_{\phi }\mathrm{/}\sqrt{r}{\mathrm{\parallel }}_{L_2({\mathrm{R}}^3\times (0,T))},$ where $v_r$ is the radial component of velocity and ${\omega }_{\phi }$ is the angular component of vorticity, imply regularity of weak solutions.