Sharp bounds on the Nusselt number in Rayleigh-Bénard convection and a bilinear estimate by Coifman-Meyer

We prove a conjecture in fluid dynamics concerning optimal bounds for heat transportation in the infinite Prandtl number limit and for large Rayleigh number, 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 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.