Measure valued solutions to the spatially homogeneous Boltzmann equation without angular cutoff

TL;DR

This paper introduces a uniform method using Toscani metric to establish the existence of measure-valued solutions to the spatially homogeneous Boltzmann equation without angular cutoff, covering both hard and soft potentials.

Contribution

It provides a novel approach to prove existence of solutions in Schwartz space without extra moment conditions for hard potentials and with bounded moments for soft potentials.

Findings

Solutions are in Schwartz space for non-Dirac initial data.

Existence results hold for both finite and infinite energy cases.

Method applies under non-angular cutoff assumptions.

Abstract

A uniform approach is introduced to study the existence of measure valued solutions to the homogeneous Boltzmann equation for both hard potential with finite energy, and soft potential with finite or infinite energy, by using Toscani metric. Under the non-angular cutoff assumption on the cross-section, the solutions obtained are shown to be in the Schwartz space in the velocity variable as long as the initial data is not a single Dirac mass without any extra moment condition for hard potential, and with the boundedness on moments of any order for soft potential.