The dissipative linear Boltzmann equation for hard spheres

TL;DR

This paper proves the existence, uniqueness, and convergence to a non-equilibrium Maxwellian equilibrium for a dissipative linear Boltzmann equation modeling inelastic collisions of gas particles with a fixed background, highlighting the impact of inelasticity on equilibrium temperature.

Contribution

It establishes the existence and uniqueness of a universal Maxwellian equilibrium with a lower temperature in a dissipative setting, and proves convergence of solutions to this equilibrium.

Findings

Existence and uniqueness of equilibrium state with unit mass.

Equilibrium is a Maxwellian with lower temperature than background.

Solutions strongly converge to equilibrium over time.

Abstract

We prove the existence and uniqueness of an equilibrium state with unit mass to the dissipative linear Boltzmann equation with hard--spheres collision kernel describing inelastic interactions of a gas particles with a fixed background. The equilibrium state is a universal Maxwellian distribution function with the same velocity as field particles and with a non--zero temperature lower than the background one, which depends on the details of the binary collision. Thanks to the H--theorem we then prove strong convergence of the solution to the Boltzmann equation towards the equilibrium.