Upper bounds on Nusselt number at finite Prandtl number

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We study Rayleigh Bénard convection based on the Boussinesq approximation. We are interested in upper bounds on the Nusselt number Nu, the upwards heat transport, in terms of the Rayleigh number Ra, that characterizes the relative strength of the driving mechanism and the Prandtl number Pr, that characterizes the strength of the inertial effects. We show that, up to logarithmic corrections, the upper bound Nu Ra¹/³ of Constantin and Doering in 1999 persists as long as Pr Ra¹/³ and then crosses over to Nu⁻¹/²Ra¹/². This result improves the one of Wang by going beyond the perturbative regime Pr ≫ Ra. The proof uses a new way to estimate the transport nonlinearity in the Navier-Stokes equations capitalizing on the no-slip boundary condition. It relies on a new Calderón-Zygmund estimate for the non-stationary Stokes equations in L¹ with a borderline Muckenhoupt weight.