A regularity criterion for the angular component of velocity in the norm $L_{\mathrm{\infty }}(0,T;L_p(\mathrm{\Omega })),\frac{3}{p}<1$ in axisymmetric 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 $\omega =\text{curl}v$ the vorticity field. We use $H^3$ Sobolev estimates for the modified stream function (stream function divided by radius) and energy type estimates for gradient of swirl to derive two order reduction estimates. Using the estimate $$\mathrm{\parallel }{v}_{\phi }{\mathrm{\parallel }}_{L_{\mathrm{\infty }}(0,T;L_p)\mathrm{\Omega })}\text{les}A,$$ where A is a given number and $p>3$ we prove the existence of global regular axially-symmetric solutions.