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 $\|{v_{r}}/{\sqrt{r^3}}\|_{L_2({\rm R}^3\times (0,T))}$ as well as boundedness of $\|{\omega_{\varphi}}/{\sqrt{r}}\|_{L_2({\rm R}^3\times (0,T))},$ where $v_r$ is the radial component of velocity and $\omega_{\varphi}$ is the angular component of vorticity, imply regularity of weak solutions.