Dipartimento di Scienze di Base e Applicate per l'Ingegneria

 

 

 

We are interested in Rayleigh-Benard convection, by which we understand the motion of a liquid in a container that is heated through the bottom and cooled through the top surface. In the Boussinesq approximation, this leads to the Navier-Stokes equations for the (divergence-free) velocity with no-slip boundary conditions coupled to an advection-diffusion equation for the temperature with inhomogeneous Dirichlet boundary conditions. The coupled system contains two nondimensional parameters: The Rayleigh Number $Ra$, that measures the strength of the imposed temperature gradient, and the Prandtl number $Pr$, that measures the strength of viscosity over inertia. We are interested in the regime of $Ra\ll 1$, in which case the fluid motion is turbulent and the temperature features sharp boundary layers. One relevant way of measuring the turbulent transport is to monitor the Nusselt number $Nu$, which is the time and space-averaged upwards heat flux. Many (expensive) experiments and (large scale) numerical simulations display several scaling regimes for $Nu$ in terms of $Ra$ and $Pr$.

 

It is very surprising that rigorous PDE theory in form of a priori estimates can contribute to the understanding of these scaling regimes: In 1999, P. Constantin and C. Doering rigorously established the upper bound $Nu\lesssim Ra^{1/3}$ (up to logarithms) in the regime of vanishing inertia, that is, for $Pr=\infty$, in which case the Navier-Stokes equation is replaced by the quasi-static Stokes equation. This upper bound is consistent with the experimental and numerical data.

 

We present an extension to finite Prandlt Number (i.e. replacing the quasi-stationary Stokes by the time-dependent Navier-Stokes equation): We show that the upper bound $Nu\lesssim Ra^{1/3}$ persists as long as $Pr\ll Ra^{1/3}$, which goes beyond the small-data regime for Navier-Stokes.

 

The proof relies on a simple but curious estimate of the transport nonlinearity in terms of the dissipation rate. This estimate naturally leads to the $L^1$-norm with a singular weight depending on the distance to the no-slip boundary. Hence we need to develop a fairly involved maximal regularity theory for the instationary Stokes equation with no-slip boundary condition with respect to this norm. This is joint work with A. Choffrut and C. Nobili.