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

Abstract

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.