MiS Preprint Repository

We consider Rayleigh--Bénard convection as modelled by the Boussinesq equations in the infinite-Prandtl-number limit. We are interested in the scaling of the average upward heat transport, the Nusselt number $Nu$, in terms of the non-dimensionalized temperature forcing, the Rayleigh number $Ra$. Experiments, asymptotics and heuristics suggest that $Nu\sim R{a}^{1/3}$.

This work is mostly inspired by two earlier rigorous work on upper bounds of $Nu$ in terms of $Ra$:
1.) The work of Constantin and Doering establishing $Nu\lesssim R{a}^{1/3}{\ln }^{2/3}Ra$ with help of a (logarithmically failing) maximal regularity estimate in $L^{\mathrm{\infty }}$ on the level of the Stokes equation.
2.) The work of Doering, Reznikoff and the first author establishing $Nu\lesssim R{a}^{1/3}{\ln }^{1/3}Ra$ with help of the background temperature method.

The paper contains two results:
1.) The background temperature method can be slightly modified to yield $Nu\lesssim R{a}^{1/3}{\ln }^{1/15}Ra$ .
2.) The estimates behind the temperature background method can be combined with the maximal regularity in $L^{\mathrm{\infty }}$ to yield $Nu\lesssim R{a}^{1/3}{\ln }^{1/3}\ln Ra$ --- an estimate that is only a double logarithm away from the supposedly optimal scaling.