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

Understand

We prove the immediate appearance of a lower bound for 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 continuity away from the grazing set of the solution.

• These results are entirely constructive if the domain is $C^3$ and strictly convex.
• 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 \textit{a priori} bounds on the solution.