Free cooling and high-energy tails of granular gases with variable restitution coefficient

TL;DR

This paper establishes the optimal algebraic cooling rate of granular gases with variable restitution coefficients, extending previous results to more realistic models and deriving new tail behavior theorems.

Contribution

It generalizes the analysis of granular gases to variable restitution coefficients, proving generalized Haff's law and establishing L1-exponential tail behavior.

Findings

Proves generalized Haff's law for variable restitution coefficients

Extends Boltzmann collision operator results to broader class of coefficients

Derives L1-exponential tail theorem for the model

Abstract

We prove the so-called generalized Haff's law yielding the optimal algebraic cooling rate of the temperature of a granular gas described by the homogeneous Boltzmann equation for inelastic interactions with non constant restitution coefficient. Our analysis is carried through a careful study of the infinite system of moments of the solution to the Boltzmann equation for granular gases and precise Lp estimates in the selfsimilar variables. In the process, we generalize several results on the Boltzmann collision operator obtained recently for homogeneous granular gases with constant restitution coefficient to a broader class of physical restitution coefficients that depend on the collision impact velocity. This generalization leads to the so-called L1-exponential tails theorem. for this model.