Global Existence Proof for Relativistic Boltzmann Equation with Hard Interactions

TL;DR

This paper proves the global existence of solutions for the relativistic Boltzmann equation with hard interactions by extending classical techniques and causality methods, under finite mass, energy, and entropy conditions.

Contribution

It extends previous nonrelativistic results to the relativistic case with hard interactions, establishing global mild solutions under broader assumptions.

Findings

Existence of global mild solutions proven

Applicable to relativistic scattering with hard interactions

Builds on DiPerna-Lions and causality techniques

Abstract

By combining the DiPerna and Lions techniques for the nonrelativistic Boltzmann equation and the Dudy\'{n}ski and Ekiel-Je\.{z}ewska device of the causality of the relativistic Boltzmann equation, it is shown that there exists a global mild solution to the Cauchy problem for the relativistic Boltzmann equation with the assumptions of the relativistic scattering cross section including some relativistic hard interactions and the initial data satisfying finite mass, energy and entropy. This is in fact an extension of the result of Dudy\'{n}ski and Ekiel-Je\.{z}ewska to the case of the relativistic Boltzmann equation with hard interactions.