The Newtonian limit of the relativistic Boltzmann equation

TL;DR

This paper proves that solutions of the relativistic Boltzmann equation converge to those of the classical Boltzmann equation as the speed of light tends to infinity, establishing a rigorous Newtonian limit.

Contribution

It provides a local existence and uniqueness theorem for the relativistic Boltzmann equation uniform in the speed of light and demonstrates the convergence to the classical equation in the non-relativistic limit.

Findings

Established local existence and uniqueness independent of c

Proved convergence to classical Boltzmann solutions as c→∞

Identified conditions for the Newtonian limit

Abstract

The relativistic Boltzmann equation for a constant differential cross section and with periodic boundary conditions is considered. The speed of light appears as a parameter $c>c_0$ for a properly large and positive $c_0$. A local existence and uniqueness theorem is proved in an interval of time independent of $c>c_0$ and conditions are given such that in the limit $c\to +\infty$ the solutions converge, in a suitable norm, to the solutions of the non-relativistic Boltzmann equation for hard spheres.