Global regular solutions to the Navier-Stokes equations in an axially symmetric domain

Abstract

We prove the existence of global regular solutions to the Navier-Stokes
equations in an axially symmetric domain in $\mathbb R^3$ and with boundary slip
conditions. We assume that initial angular component of velocity and angular
component of the external force and angular derivatives of the cylindrical
components of initial velocity and of the external force are sufficiently
small in corresponding norms. Then there exists a solution such that velocity
belongs to $W_{5/2}^{2,1}(\Omega^T)$ and gradient of pressure to
$L_{5/2}(\Omega^T)$, and we do not have restrictions on $T$.