SOLUTIONS TO THE BOLTZMANN-EQUATION IN THE BOUSSINESQ REGIME

We consider a gas in a horizontal slab in which the top and bottom wal ls 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 K nudsen 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 exist s a solution to the BE for t is an element of(0, t) which is near a gl obal Maxwellian, and whose moments are close, up to order epsilon(2), to the density, velocity and temperature obtained from the smooth solu tion of the Oberbeck-Boussinesqequations assumed to exist for t less t han or equal to (t) over bar.