Friday, February 8, 2013

1302.1755 (Marc Briant)

Instantaneous filling of the vacuum for the full Boltzmann equation in
bounded domains

Marc Briant
We prove the immediate appearance of an explicit lower bound for continuous mild solutions to the full Boltzmann equation in the torus or a $C^2$ convex domain with specular boundary conditions, under the sole assumption of regularity of the solution. We investigate a wide range of collision kernels, some satisfying Grad's cutoff assumption and others not. We show that this lower bound is exponential, independent of time and space with explicit constants depending only on the a priori bounds on the solution. In particular, this lower bound is Maxwellian in the case of cutoff collision kernels. A thorough study of characteristic trajectories, as well as a geometric approach of grazing collisions against the boundary are derived.
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