Moments creation for the inelastic Boltzmann equation for hard potentials without angular cutoff

TL;DR

This paper proves the global existence of measure-valued solutions for the inelastic Boltzmann equation with hard potentials without angular cutoff and demonstrates the creation of polynomial moments, highlighting a key property of such solutions.

Contribution

It establishes the existence of solutions and proves polynomial moment creation for the inelastic Boltzmann equation with hard potentials, without requiring entropy bounds.

Findings

Global-in-time existence of measure-valued solutions

Creation of polynomial moments for solutions

Refined Povzner-type inequality for inelastic collisions

Abstract

This paper is concerned with the inelastic Boltzmann equation without angular cutoff. We work in the spatially homogeneous case. We establish the global-in-time existence of measure-valued solutions under the generic hard potential long-range interaction on the collision kernel. In addition, we provide a rigorous proof for the creation of polynomial moments of the measure-valued solutions, which is a special property that can only be expected from hard potential collisional cross-sections. The proofs rely crucially on the establishment of a refined Povzner-type inequality for the inelastic Boltzmann equation without angular cutoff. The class of initial data that we require is general in the sense that we only require the boundedness of $(2+\kappa)$-moment for $\kappa>0$ and do not assume any entropy bound.