Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation

TL;DR

This paper proves that solutions to the spatially homogeneous Boltzmann equation with cutoff hard potentials stay uniformly bounded above by a Maxwellian distribution over time, given initial bounds, using a comparison principle.

Contribution

It introduces a novel comparison principle based on dissipative properties to establish uniform upper Maxwellian bounds for solutions.

Findings

Solutions remain bounded by Maxwellian distributions over time

The technique applies to both homogeneous and inhomogeneous cases

Provides insights into propagation of bounds in kinetic equations

Abstract

For the spatially homogeneous Boltzmann equation with cutoff hard potentials it is shown that solutions remain bounded from above, uniformly in time, by a Maxwellian distribution, provided the initial data have a Maxwellian upper bound. The main technique is based on a comparison principle that uses a certain dissipative property of the linear Boltzmann equation. Implications of the technique to propagation of upper Maxwellian bounds in the spatially-inhomogeneous case are discussed.