We prove a conjecture in fluid dynamics concerning optimal bounds for heat transportation in the infinite Prandtl number limit and for large Rayleigh number $\mathrm{Ra}$ , predicted in (Howard in Proceedings of the 11th International Congress of Applied Mathematics on Applied Mechanics, Munich, 1964, p. 1109, Springer, 1966) and (Malkus in Proc. R. Soc. Lond. Ser. A 225:196–212, 1954). Due to a maximum principle property for the temperature exploited by Constantin-Doering and Otto-Seis, this amounts to showing a-priori bounds for horizontally-periodic solutions of a fourth-order equation in a strip of large width. While there have been recent nearly-optimal results up to logarithmic divergences in $\mathrm{Ra}$ , we prove here sharp bounds employing Fourier analysis, integral representations, and a bilinear estimate due to Coifman and Meyer which uses the Carleson measure characterization of BMO functions by Fefferman.
