  In this paper, we study the three-dimensional axisymmetric Navier-Stokes
system with nonzero swirl. By establishing a new key inequality for the pair
$(\frac{\omega^{r}}{r},\frac{\omega^{\theta}}{r})$, we get several Prodi-Serrin
type regularity criteria based on the angular velocity, $u^\theta$. Moreover,
we obtain the global well-posedness result if the initial angular velocity
$u_{0}^{\theta}$ is appropriate small in the critical space $L^{3}(\R^{3})$.
Furthermore, we also get several Prodi-Serrin type regularity criteria based on
one component of the solutions, say $\omega^3$ or $u^3$.
