Stability of two-dimensional Navier-Stokes motions in the periodic case

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.