Some stability problem for the Navier–Stokes equations in the periodic case

The Navier–Stokes motions in a box with periodic boundary conditions are considered. First the existence of global regular two-dimensional solutions is proved. Since the external force does not decay in time, the solution has the same property. The necessary estimates and existence are proved step by step in time. Dissipation in the Navier–Stokes equations makes this approach possible. Assuming that the initial velocity and the external force are sufficiently close to the initial velocity and the external force of the two-dimensional problems we prove existence of global three-dimensional regular solutions which remain close to the two-dimensional solutions for all time.