A regularity criterion for the angular component of velocity in the norm $L_{\infty }(0,T;L_p(\hat{I}\text{c◯})),\frac{3}{p}<1$ in axisymmetric Navier Stokes equations in a cylinder

Abstract

We consider the axisymmetric Navier-Stokes equations in a finite cylinder $\hat{I}\text{c◯}\subset {\mathbb{R}}^3$. We assume that $v_r$, $v_{\ddot{I}}†$, $\ddot{I}{‰}_{\ddot{I}}†$ vanish on the lateral part of boundary $\partial \hat{I}\text{c◯}$ of the cylinder, and that $v_z$, $\ddot{I}{‰}_{\ddot{I}}†$, ${\partial }_z{v}_{\ddot{I}}†$ vanish on the top and bottom parts of the boundary $\partial \hat{I}\text{c◯}$, where we used standard cylindrical coordinates, and we denoted by $ω=\curl v$ 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 \[ \|v_φ\|_{L_\infty(0,T;L_p)Ω)}\les A, \] where A is a given number and $p>3$ we prove the existence of global regular axially-symmetric solutions.