Stability for Rayleigh-Benard convective solutions of the Boltzmann equation

TL;DR

This paper proves the stability of a 2D convective solution to the Boltzmann equation in a gravitational setting, near the bifurcation point, for small Knudsen number and small perturbations.

Contribution

It establishes the stability of a convective solution to the Boltzmann equation close to the Oberbeck-Boussinesq solution near bifurcation, for small Knudsen number.

Findings

Stability of the convective solution is proven near the bifurcation point.

The solution remains stable under small 2D perturbations.

Results apply for Rayleigh numbers close to the bifurcation point.

Abstract

We consider the Boltzmann equation for a gas in a horizontal slab, subject to a gravitational force. The boundary conditions are of diffusive type, specifying the wall temperatures, so that the top temperature is lower than the bottom one (Benard setup). We consider a 2-dimensional convective stationary solution, which is close for small Knudsen number to the convective stationary solution of the Oberbeck-Boussinesq equations, near above the bifurcation point, and prove its stability under 2-d small perturbations, for Rayleigh number above and close to the bifurcation point and for small Knudsen number.