Mathematical treatment of the homogeneous Boltzmann equation for Maxwellian molecules in the presence of singular kernels

TL;DR

This paper establishes the existence of weak solutions to the homogeneous Boltzmann equation for Maxwellian molecules with singular kernels, under minimal regularity assumptions, and proves conservation laws without entropy bounds.

Contribution

It provides a mathematical proof of weak solutions for the Boltzmann equation with very weak cutoff conditions and without entropy boundedness.

Findings

Existence of weak solutions under weak cutoff conditions

Conservation of momentum and energy proven

Solutions include initial data from all Borel probability measures with finite second moments

Abstract

This paper proves the existence of weak solutions to the spatially homogeneous Boltzmann equation for Maxwellian molecules, when the initial data are chosen from the space of all Borel probability measures on R^3 with finite second moments and the (angular) collision kernel satisfies a very weak cutoff condition. Conservation of momentum and energy is also proved for these weak solutions, without resorting to any boundedness of the entropy.