The Oberbeck–Boussinesq approximation and Rayleigh–Bénard convection revisited

We consider the Oberbeck–Boussinesq approximation driven by an inhomogeneous temperature distribution on the boundary of a bounded fluid domain. The relevant boundary conditions are perturbed by a non–local term arising in the incompressible limit of the Navier–Stokes–Fourier system. The long time behaviour of the resulting initial/boundary value problem is investigated.