Existence and Homogenization of the Rayleigh-Bénard Problem

The Navier-Stokes equation driven by heat conduction is studied. As a prototype we consider Rayleigh-Bénard convection, in the Boussinesq approximation. Under a large aspect ratio assumption, which is the case in Rayleigh-Bénard experiments with Prandtl number close to one, we prove the existence of a global strong solution to the 3D NavierStokes equation coupled with a heat equation, and the existence of a maximal B-attractor. A rigorous two-scale limit is obtained by homogenization theory. The mean velocity field is obtained by averaging the two-scale limit over the unit torus in the local variable. PACS numbers, 44.25.+f, 47.27.Te and The University of Iceland, Science Institute, Dunhaga, Reykjavík 107, Iceland. †Email: [email protected] and URL: www.math.ucsb.edu/ ̃birnir ‡and Department of Mathematics, Chalmers University of Technology and Göteborg University S-412 96 Göteborg, Sweden. §Email: [email protected] and URL: www.math.chalmers.se/ ̃nilss 1