Bounds on the Nusselt Number for Marangoni Convection

We use the background method to prove rigorous upper bounds on the Nusselt number in terms of the Marangoni number in Marangoni convection. When the Prandtl number is infinite $Nu \leq .84 Ma^{2/7}$. For finite Prandtl number we proved that $Nu \la Ma^{1/2}$. We compare these to numerical simulations by Boeck and Thess that suggest that for real flows $Nu\la Ma^{2/9}$. We also use the background method and non-variational techniques to improve the lower bound for the critical Marangoni number for energy stability of the conduction solution in the infinite Prandtl number case.