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

Abstract

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.