Sharp bounds on the Nusselt number in Rayleigh-Bénard convection and a bilinear estimate via Carleson measures
Sagun Chanillo, Andrea Malchiodi
TL;DR
This work proves a sharp universal bound on heat transport in Rayleigh-Bénard convection at infinite Prandtl number, establishing $Nu \le C_0$ as $H \to \infty$, which corresponds to $Nu_{\text{phys}} \le C_0 (\text{Ra})^{1/3}$ in the standard scaling. The authors combine a temperature maximum principle with a fourth-order reformulation, applying Fourier analysis, integral representations, and a bilinear Coifman–Meyer estimate grounded in Carleson-measure theory to control horizontal-averaged interactions. A key step uses the bi-Laplacian Green’s function in the upper half-space to represent the solution, decomposes the source into bottom/top contributions, and employs Carleson-measure–based paraproduct bounds to bound $\langle \theta w \rangle$ uniformly, followed by a bi-harmonic boundary-correction to fix the top boundary. The result advances a rigorous $Ra^{1/3}$-type bound for heat transport in high-Rayleigh-number convection and showcases a framework that integrates harmonic analysis with PDE boundary-value techniques for convection problems.
Abstract
We prove a conjecture in fluid dynamics concerning optimal bounds for heat transportation in the infinite Prandtl number limit. Due to a maximum principle property for the temperature exploited by Constantin-Doering and Otto-Seis, this amounts to proving a-priori bounds for horizontally-periodic solutions of a fourth-order equation in a strip of large width. Such bounds are obtained here using Fourier analysis, integral representations, and a bilinear estimate due to Coifman and Meyer which uses the Carleson measure characterization of BMO functions by Fefferman.