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Solutions to the Boltzmann equation in the Boussinesq regime

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 $ε$ is small and the temperature difference between the walls as well as the velocity field is of order $ε$, while the gravitational force is of order $ε^2$. 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 $ε^2$ to the density, velocity and temperature obtained from the smooth solution of the Oberbeck-Boussinesq equations, up to the time this one stays regular.