  In this paper we consider, by means of a precise spectral analysis, the 3D
Navier-Stokes equations endowed with Navier slip-with-friction boundary
conditions. We study the problem in a very simple geometric situation as the
region between two parallel planes, with periodicity along the two planes. This
setting, which is often used in the theory of boundary layers, requires some
special treatment for what concerns the functional setting and allows us to
characterize in a rather explicit manner eigenvalues and eigenfunctions of the
associated Stokes problem. These, will be then used in order to identify
infinite dimensional classes of data leading to global strong solutions for the
corresponding evolution Navier-Stokes equations.
