Bounds on vertical heat transport for infinite Prandtl number Rayleigh-B\'enard convection

For the infinite Pandtl number limit of the Boussineq equations, the enhancement of vertical heat transport in Rayleigh-B\'enard convection, the Nusselt number $Nu$, is bounded above in terms of the Rayleigh number $Ra$ according to $Nu \leq .644 \times Ra^{1/3} [\log Ra]^{1/3}$ as $Ra \rightarrow \infty$. This rigorous estimate follows from the utilization of a novel logarithmic profile in the background method for producing bounds on bulk transport together with new estimates for the bi-Laplacian in a weighted $L^2$ space. It is a quantitative improvement over the best currently available analytic result, and it comes within the mild logarithmic factor of the pure 1/3 scaling anticipated by both the classical marginally stable boundary layer argument and the most recent high-resolution numerical computations of the optimal bound on $Nu$ using the background method.