Infinite energy solutions to the homogeneous Boltzmann equation

TL;DR

This paper develops a method to construct unique infinite energy solutions to the homogeneous Boltzmann equation for Maxwellian molecules, analyzing their long-term behavior and stability without requiring finite second moments.

Contribution

It introduces a novel approach to solving the homogeneous Boltzmann equation in a space that includes infinite energy solutions, extending previous methods.

Findings

Constructed unique solutions in a space allowing infinite energy

Analyzed large time asymptotics of solutions

Provided an elementary proof of stability for self-similar solutions

Abstract

The goal of this work is to present an approach to the homogeneous Boltzmann equation for Maxwellian molecules with a physical collision kernel which allows us to construct unique solutions to the initial value problem in a space of probability measures defined via the Fourier transform. In that space, the second moment of a measure is not assumed to be finite, so infinite energy solutions are not {\it a priori} excluded from our considerations. Moreover, we study the large time asymptotics of solutions and, in a particular case, we give an elementary proof of the asymptotic stability of self-similar solutions obtained by A.V. Bobylev and C. Cercignani [J. Stat. Phys. {\bf 106} (2002), 1039--1071].