  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
$\epsilon$ is small and the temperature difference between the walls as well as
the velocity field is of order $\epsilon$, while the gravitational force is of
order $\epsilon^2$. We prove that there exists a solution to the BE for which
is near a global Maxwellian, and whose moments are close, up to order
$\epsilon^2$ to the density, velocity and temperature obtained from the smooth
solution of the Oberbeck-Boussinesq equations, up to the time this one stays
regular.
