A new characterization and global regularity of infinite energy solutions to the homogeneous Boltzmann equation

TL;DR

This paper introduces a new way to characterize infinite energy solutions to the homogeneous Boltzmann equation, improving previous results and establishing their global regularity and smoothing effects.

Contribution

It provides a complete characterization of infinite energy solutions for Maxwellian cross sections, enhancing the understanding of measure-valued solutions.

Findings

New characterization of characteristic functions capturing physical moment constraints

Complete description of infinite energy solutions for Maxwellian cross section

Proven global smoothing effect for these solutions except for a single Dirac mass

Abstract

The purpose of this paper is to introduce a new characterization of the characteristic functions for the study on the measure valued solution to the homogeneous Boltzmann equation so that it precisely captures the moment constraint in physics. This significantly improves the previous result by Cannone-Karch [CPAM 63(2010), 747-778] in the sense that the new characterization gives a complete description of infinite energy solutions for the Maxwellian cross section. In addition, the global in time smoothing effect of the infinite energy solution except for a single Dirac mass initial datum is justified as for the finite energy solution.