Two proofs of Haff's law for dissipative gases: the use of entropy and the weakly inelastic regime

TL;DR

This paper provides two simplified proofs of Haff's law for granular gases, one using entropy and the other based on moments, applicable under different conditions of inelasticity and initial data.

Contribution

It introduces two new, more straightforward proofs of Haff's law, expanding understanding of dissipative gas behavior with variable restitution coefficients.

Findings

Haff's law holds for initial data with finite entropy.

Haff's law is valid in a weakly inelastic regime for non-constant restitution.

The proofs simplify previous approaches to granular gas analysis.

Abstract

We revisit our recent contribution (SIAM J. Math. Analysis, 42 (2010) 2499--2538) and give two simpler proofs of the so-called Haff's law for granular gases (with non-necessarily constant restitution coefficient). The first proof is based upon the use of entropy and asserts that Haff's law holds whenever the initial datum is of finite entropy. The second proof uses only the moments of the solutions and holds in some weakly inelasticity regime which has to be clearly defined whenever the restitution coefficient is non-constant