The initial layer problem and infinite Prandtl number limit of Rayleigh-Benard convection

In this paper the Initial layer problem and infinite Prandtl number limit of Rayleigh-Benard convection are studied. For the case of ill-prepared initial data infinite Prandtl number limit of the Boussinesq approximation for Rayleigh-Benard convection is proven by using the asymptotic expansion methods of singular perturbation theory and the classical energy methods. An exact approximating solution with the zero order term and the 1st order term expansion is given and the convergence rates O(epsilon(3/2)) and O(epsilon(2)) are respectively obtained. This improves the result of X. M. Wang [Commun. Pure Appli. Math., LVII(2004), 1265-1282].