On the Measure Valued Solution to the Inelastic Boltzmann Equation with Soft Potentials

TL;DR

This paper extends the existence and uniqueness of measure-valued solutions to the inelastic Boltzmann equation with soft potentials, including non-cutoff cases, and analyzes their moment propagation and energy dissipation.

Contribution

It generalizes previous results from elastic to inelastic interactions for soft potentials, covering both cutoff and non-cutoff cases with finite and infinite energy data.

Findings

Existence and uniqueness of measure-valued solutions established.

Proved solutions propagate moments and dissipate energy.

Extended results to non-cutoff inelastic Boltzmann equations.

Abstract

The goal of this paper is to extend the existence result of measure-valued solution to the Boltzmann equation in elastic interaction, given by Morimoto-Wang-Yang, to the inelastic Boltzmann equation with moderately soft potentials, also as an extensive work of our preceding result in the inelastic Maxwellian molecules case. We prove the existence and uniqueness of measure-valued solution under Grad's angular cutoff assumption, as well as the existence of non-cutoff solution, for both finite and infinite energy initial datum, by a delicate compactness argument. In addition, the moments propagation and energy dissipation properties are justified for the obtained measure-valued solution as well.