Sharp Bounds for the Nusselt number in Rayleigh-Benard Convection

Abstract: We present in this talk a resolution of a 65 year old conjecture in Fluid Mechanics due to Malkus. Thermal convection processes can be modeled by what is called the Rayleigh-Benard model. In this talk we consider the Boussinesq approximation to the Rayleigh-Benard model  in the infinite Prandtl number limit. Two fundamental constants associated with this model is the Rayleigh number and the Nusselt number which is a measure of the ratio of the total heat flux to the heat flux via conduction in the absence of fluid  flow. We obtain sharp bounds for the Nusselt number by the Rayleigh number. A large body of numerical and experimental evidence supported the conjecture. Previous mathematically rigorous results were obtained in this direction by P. Constantin-C. Doering and F. Otto-C. Seis.  This is joint work with Andrea Malchiodi.