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

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 in (J Stat Phys 165:866-906, 2016), to the inelastic Boltzmann equation with moderately soft potentials, which is also an extensive work of our preceding result in the inelastic Maxwellian molecules case. We first prove the existence and uniqueness of measure-valued solution under Grad's angular cutoff assumption, based on which, we further obtain 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.