  We consider the motion described by the Navier-Stokes equations in a box with
periodic boundary conditions. First we prove the existence of global strong
two-dimensional solutions. Next we show the existence of global strong
three-dimensional solutions under the assumption that the initial data and the
external force are sufficiently close to the initial data and the external
force of the two-dimensional problem in appropriate spaces. The second result
can be treated as stability of strong two-dimensional solutions in the set of
suitably strong three-dimensional motions.
