New bounds on the vertical heat transport for B\'enard-Marangoni convection at infinite Prandtl number

We prove a new rigorous upper bound on the vertical heat transport for B'enard-Marangoni convection of a two- or three-dimensional fluid layer with infinite Prandtl number. Precisely, for Marangoni number $Ma\gg 1$ the Nusselt number $Nu$ is bounded asymptotically by $Nu\lesssim M{a}^{2\mathrm{/}7}(\ln Ma{)}^{-1\mathrm{/}7}$. Key to our proof are a background temperature field with a hyperbolic profile near the fluid’s surface, and new estimates for the coupling between temperature and vertical velocity.