On Measure Solutions of the Boltzmann Equation, part I: Moment Production and Stability Estimates

TL;DR

This paper establishes the existence, moment production, and stability of measure-valued solutions to the spatially homogeneous Boltzmann equation with hard potentials, including cases with and without angular cutoff.

Contribution

It introduces a new approach to prove existence of weak measure solutions using the Mehler transform and provides moment and stability estimates for these solutions.

Findings

Existence of weak measure solutions for the Boltzmann equation with hard potentials.

Moment production estimates in standard and exponential forms.

Uniqueness and strong stability results under Grad angular cutoff.

Abstract

The spatially homogeneous Boltzmann equation with hard potentials is considered for measure valued initial data having finite mass and energy. We prove the existence of \emph{weak measure solutions}, with and without angular cutoff on the collision kernel; the proof in particular makes use of an approximation argument based on the Mehler transform. Moment production estimates in the usual form and in the exponential form are obtained for these solutions. Finally for the Grad angular cutoff, we also establish uniqueness and strong stability estimate on these solutions.