Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions

TL;DR

This paper establishes an immediate exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions, applicable to a wide range of collision kernels, including Maxwellian in the cutoff case.

Contribution

It provides the first proof of uniform-in-time exponential lower bounds for solutions to the Boltzmann equation under Maxwellian diffusion boundary conditions, covering various collision kernels.

Findings

Exponential lower bounds are uniform in time and space.

Results are constructive if initial data has no vacuum.

Lower bounds are Maxwellian for cutoff collision kernels.

Abstract

We prove the immediate appearance of an exponential lower bound, uniform in time and space, for continuous mild solutions to the full Boltzmann equation in a $C^2$ convex bounded domain with the physical Maxwellian diffusion boundary conditions, under the sole assumption of regularity of the solution. We investigate a wide range of collision kernels, with and without Grad's angular cutoff assumption. In particular, the lower bound is proven to be Maxwellian in the case of cutoff collision kernels. Moreover, these results are entirely constructive if the initial distribution contains no vacuum, with explicit constants depending only on the \textit{a priori} bounds on the solution.