Large time regular solutions to the Navier-Stokes equations in cylindrical domains

We prove the large time existence of solutions to the Navier-Stokes equations with slip boundary conditions in a cylindrical domain. Assuming smallness of $L_2$-norms of derivatives of initial velocity with respect to variable along the axis of the cylinder, we are able to obtain estimate for velocity in $W^{2,1}_2$ without restriction on its magnitude. Then existence follows from the Leray-Schauder fixed point theorem