Moment inequalities and high-energy tails for the Boltzmann equations with inelastic interactions

TL;DR

This paper investigates the high-energy behavior of steady-state velocity distributions in granular media modeled by inelastic Boltzmann equations, establishing exponential tail estimates that depend on external forcing models.

Contribution

It provides integral estimates for solutions of inelastic Boltzmann equations, revealing how tail behavior varies with different external forcing regimes.

Findings

Velocity distributions decay as $C ext{exp}(-r|v|^s)$ for large $|v|$

Tail order $s$ ranges from 1 to 2 depending on external forcing

Method uses moment inequalities and Povzner-type estimates

Abstract

We study the high-energy asymptotics of the steady velocity distributions for model systems of granular media in various regimes. The main results obtained are integral estimates of solutions of the hard-sphere Boltzmann equations, which imply that the velocity distribution functions $f(v)$ behave in a certain sense as $C\exp(-r|v|^s)$ for $|v|$ large. The values of $s$, which we call {\em the orders of tails}, range from $s=1$ to $s=2$, depending on the model of external forcing. The method we use is based on the moment inequalities and careful estimating of constants in the integral form of the Povzner-type inequalities.