On the regularity of axially-symmetric solutions to the incompressible 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 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. We use weighted estimates and $H^3$ Sobolev estimate on the modified stream function to derive three order-reduction estimates. These enable one to reduce the order of the nonlinear estimates of the equations, and help observe that the solutions to the equations are ``almost regular’'. We use the order-reduction estimates to show that the solution to the equations remains regular as long as, for any $p\in (6,\mathrm{\infty })$, $\mathrm{\parallel }{v}_{\phi }{\mathrm{\parallel }}_{L_t^{\mathrm{\infty }}{L}_x^p}\mathrm{/}\mathrm{\parallel }{v}_{\phi }{\mathrm{\parallel }}_{L_t^{\mathrm{\infty }}{L}_x^{\mathrm{\infty }}}$ remains bounded below by a positive number.