Stability of two-dimensional solutions to the Navier-Stokes equations in cylindrical domains under Navier-boundary conditions

Abstract

The Navier-Stokes motions in cylindrical domain with Navier boundary conditions are considered. First the existence of global regular two-dimensional solutions are proved. The solutions are bounded by the same constant for all time. 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 global existence of three-dimensional solutions which remain close to the two-dimensional solutions for all time. In this way we mean stability of two-dimensional solutions.