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| We consider Rayleigh-B\'{e}nard 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 $Nu\sim Ra^{\frac{1}{3}}$ and $Nu\sim Ra^{\frac{1}{2}}$ 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 in $L^{\infty}$ and in $L^{1}$ for 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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