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

TL;DR

This paper proves that solutions to the full Boltzmann equation in convex domains instantly develop a positive lower bound, which is exponential and explicit, under broad conditions including various collision kernels.

Contribution

It establishes the immediate filling of the vacuum for the Boltzmann equation in convex domains with explicit bounds, covering a wide class of collision kernels and boundary conditions.

Findings

Lower bound appears instantly for solutions in convex domains.

The lower bound is exponential and independent of time and space.

Results are constructive for $C^3$ convex domains with explicit constants.

Abstract

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. 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.