  The Navier-Stokes motions in a box with periodic boundary conditions are
considered. First the existence of global regular two-dimensional solutions is
proved. The solutions are such that continuous with respect to time norms are
controlled by the same constant for all $t\in\mathbb{R}_+$. Assuming that the
initial velocity and the external force are sufficiently close to the initial
velocity and the external force of the two-dimensional solutions we prove
existence of global three-dimensional regular solutions which remain close to
the two-dimensional solutions for all time. In this way we mean stability of
two-dimensional solutions.
