A Regularity Criterion for the Angular Component of Velocity in the norm $$L_q(0,T;L_p(\Omega )),\;\frac{3}{p} +\frac{2}{q}<1,\;q< \infty $$, in Axisymmetric Navier-Stokes Equations in a Cylinder

We consider the axisymmetric Navier-Stokes equations in a finite cylinder $$\Omega \subset \mathbb {R}^3$$ . We assume that $$v_r$$ , $$v_\varphi $$ , $$\omega _\varphi $$ vanish on the lateral part of boundary $$\partial \Omega $$ of the cylinder, and that $$v_z$$ , $$\omega _\varphi $$ , $$\partial _zv_\varphi $$ vanish on the top and bottom parts of the boundary $$\partial \Omega $$ , where we used standard cylindrical coordinates, and we denoted by $$\omega =\operatorname {curl}v$$ 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. Finally, using the estimate $$ \Vert v_\varphi \Vert _{L_q(0,T;L_p(\Omega ))} \le A, $$ where A is a given number and $$\frac{3}{p} + \frac{2}{q} < 1$$ , $$q<\infty $$ we prove the existence of global regular axially-symmetric solutions.