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

Abstract

We consider the axisymmetric Navier-Stokes equations in a finite cylinder $\Omega\subset\R^3$. 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 $\omega=\curl v$ 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 \[ \|v_\varphi\|_{L_\infty(0,T;L_p)\Omega)}\les A, \] where A is a given number and $p>3$ we prove the existence of global regular axially-symmetric solutions.