Seminar Analysis: Camilla Nobili (Max Planck Institute for Mathematics in the Sciences, Leipzig)

Rayleigh-Bénard convection at finite Prandtl number:bounds on the Nusselt number

We consider Rayleigh-Bénard convection at finite Prandtl number as modelled by the Boussinesq equation. We are interested in the scaling of the average upward heat transport, the Nusselt number Nu, in terms of the Rayleigh number Ra, and the Prandtl number Pr.

Physically motivated heuristics suggest the scaling Nu∼Ra^1⁄3 and Nu∼Ra^1/2 depending on Pr, in different regimes.

In this talk I present a rigorous upper bound for Nu reproducing both physical scalings in some parameter regimes up to logarithms. This is obtained by a (logarithmically failing) maximal regularity estimate inL1and inL1for the nonstationary Stokes equation with forcing term given by the buoyancy term and the nonlinear term, respectively. This is a joint work with Felix Otto and Antoine Choffrut.

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