Stability of two-dimensional heat-conducting incompressible motions in a cylinder

We consider the motion of an incompressible heat-conducting fluid described by the Rayleigh–Benard equations in a cylinder, where the external force depends on temperature. We assume the slip boundary conditions for the velocity and the Dirichlet condition for the temperature. First, we prove the existence of a strong global two-dimensional solution with nonvanishing in time external force. Next, we show the existence of a global three-dimensional solution to the problem assuming that its data are close to the data of the two-dimensional problem in appropriate norms. In this way we prove stability of strong two-dimensional solutions in the set of three-dimensional solutions.