On global regular solutions to the Navier-Stokes equations in cylindrical domains

We consider the incompressible fluid motion described by the Navier-Stokes equations in a cylindrical domain $\Omega\subset\R^3$ under the slip boundary conditions. First we prove long time existence of regular solutions such that $v\in W_2^{2,1}(\Omega\times(0,T))$, $\nabla p\in L_2(\Omega\times(0,T))$, where $v$ is the velocity of the fluid and $p$ the pressure. To show this we need smallness of $\|v_{,x_3}(0)\|_{L_2(\Omega)}$ and $\|f_{,x_3}\|_{L_2(\Omega\times(0,T))}$, where $f$ is the external force and $x_3$ is the axis along the cylinder. The above smallness restrictions mean that the considered solution remains close to the two-dimensional solution, which, as is well known, is regular. Having $T$ sufficiently large and imposing some decay estimates on $\|f(t)\|_{L_2(\Omega)}$ we continue the local solution step by step up to the global one