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

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 $T$