Blow up Analysis for Anomalous Granular Gases

TL;DR

This paper analyzes the long-term behavior of solutions to an energy-dependent inelastic Boltzmann equation for granular gases, focusing on finite-time cooling and blow-up phenomena in anomalous dissipative conditions.

Contribution

It introduces new self-similar variables to study blow-up profiles and generalizes classical cooling laws for anomalous granular gases.

Findings

Existence and uniqueness of blow-up profiles established.

Generalized Haff's Law for anomalous cooling derived.

Asymptotic behavior characterized with and without drift term.

Abstract

We investigate in this article the long-time behaviour of the solutions to the energy-dependant, spatially-homogeneous, inelastic Boltzmann equation for hard spheres. This model describes a diluted gas composed of hard spheres under statistical description, that dissipates energy during collisions. We assume that the gas is "anomalous", in the sense that energy dissipation increases when temperature decreases. This allows the gas to cool down in finite time. We study existence and uniqueness of blow up profiles for this model, together with the trend to equilibrium and the cooling law associated, generalizing the classical Haff's Law for granular gases. To this end, we investigate the asymptotic behaviour of the inelastic Boltzmann equation with and without drift term by introducing new strongly "nonlinear" self-similar variables.