High-velocity tails of the inelastic and the multi-species mixture Boltzmann equations

TL;DR

This paper investigates the high-velocity behavior of solutions to various homogeneous Boltzmann equations, deriving bounds and characterizations for inelastic and mixture models with different collision kernels and potentials.

Contribution

It extends existing methods to establish bounds on high-velocity tails for inelastic and mixture Boltzmann equations, including noncutoff and cutoff cases, with new exponential tail estimates.

Findings

Inelastic Boltzmann solutions have tails bounded below by exponential functions with power p between 2 and 6.213.

Mixture Boltzmann equations exhibit Maxwellian tails with p=2.

Extended mathematical lemmas enable these tail estimates.

Abstract

We study high-velocity tails of some homogeneous Boltzmann equations on $v \in \mathbb{R}_{v}^d$. First, we consider spatially homogeneous inelastic Boltzmann equation with noncutoff collision kernel, in the case of moderately soft potentials. We also study spatially homogeneous mixture Boltzmann equations : for both noncutoff collision kernel with moderately soft potentials and cutoff collision kernel with hard potentials. In the case of noncutoff inelastic Boltzmann, we obtain \[ f(t,v) \geq a(t) e^{-b(t)|v|^p}, \quad 2 < p < 6.213 \] by extending Cancellation lemma and spreading lemma and assuming $f\in C^{\infty}$. For the Mixture type Boltzmann equations, we prove Maxwellian $p=2$.