On global regular axially-symmetric solutions to the 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 $\omega= {\rm curl}\, v$ the vorticity field. Our aim is to derive the estimate $$ \left\|\frac{\omega_{r}}{r}\right\|_{V\left(\Omega\times (0,t)\right)}+\left\|\frac{\omega_{\varphi}}{r}\right\|_{V\left(\Omega\times (0,t)\right)} \leq \phi(\operatorname{data}),$$ where $\phi$ is an increasing positive function and $\|\ \|_{V\left(\Omega\times (0,t)\right)}$ is the energy norm. We are not able to derive any global type estimate for nonslip boundary conditions.