Esposito R., Lebowitz J. L.,. ,Marra R.
Solutions to the Boltzmann equation in the Boussinesq regime.
(690K, ps)
ABSTRACT. We consider a gas in a horizontal slab, in which
the top and bottom walls are kept at different temperatures. The system is described
by the Boltzmann equation (BE) with Maxwellian boundary conditions specifying the wall
temperatures. We study the behavior of the system when the Knudsen number $\e$ is
small and the temperature difference between the walls as well as the velocity field
is of order $\e$, while the gravitational force is of order $\e^2$. We prove that
there exists a solution to the BE for $t\in (0,\bar t)$ which is near a global
Maxwellian, and whose moments are close, up to order $\e^2$ to the density,
velocity and temperature obtained from the smooth solution of the Oberbeck-Boussinesq
equations assumed to exist for $t\le \bar t$.