On global regular axially-symmetric solutions to the Navier-Stokes equations in a cylinder

We consider the axisymmetric Navier-Stokes equations in a finite cylinder $\mathrm{\Omega }\subset {\mathbb{R}}^3$. We assume that $v_r$, $v_{\phi }$, ${\omega }_{\phi }$ vanish on the lateral part of boundary $\mathrm{\partial }\mathrm{\Omega }$ of the cylinder, and that $v_z$, ${\omega }_{\phi }$, ${\mathrm{\partial }}_z{v}_{\phi }$ vanish on the top and bottom parts of the boundary $\mathrm{\partial }\mathrm{\Omega }$, where we used standard cylindrical coordinates, and we denoted by $\omega =\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}v$ the vorticity field. Our aim is to derive the estimate $${\parallel \frac{{\omega }_r}{r}\parallel }_{V(\mathrm{\Omega }\times (0,t))}+{\parallel \frac{{\omega }_{\phi }}{r}\parallel }_{V(\mathrm{\Omega }\times (0,t))}\le \phi (\mathrm{data}),$$ where $\phi$ is an increasing positive function and $\mathrm{\parallel }{\mathrm{\parallel }}_{V(\mathrm{\Omega }\times (0,t))}$ is the energy norm. We are not able to derive any global type estimate for nonslip boundary conditions.