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

• 成果类型: Article
• 署名作者: Chanillo, Sagun; Malchiodi, Andrea
• 署名单位: Rutgers University System; Rutgers University New Brunswick; Scuola Normale Superiore di Pisa
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01326-z
• 发表日期: 2025
• 页码: 633-660
• 关键词: vertical heat-transport; energy-dissipation; variational bounds; FLOW

摘要

We prove a conjecture in fluid dynamics concerning optimal bounds for heat transportation in the infinite Prandtl number limit and for large Rayleigh number Ra\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathrm{Ra}$\end{document}, 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathrm{Ra}$\end{document}, 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.