Global Newtonian limit for the Relativistic Boltzmann Equation near Vacuum

TL;DR

This paper proves that solutions to the relativistic Boltzmann equation with near vacuum initial data converge to the classical Newtonian Boltzmann solutions as the speed of light tends to infinity, establishing the first global Newtonian limit result for a kinetic equation.

Contribution

It provides the first proof of the global-in-time Newtonian limit for the relativistic Boltzmann equation with near vacuum data, uniformly in the speed of light parameter.

Findings

Solutions exist globally in time uniformly in c

Convergence rate to Newtonian solutions is 1/c^{2-ε}

First proof of global Newtonian limit for a kinetic equation

Abstract

We study the Cauchy Problem for the relativistic Boltzmann equation with near Vacuum initial data. Unique global in time "mild" solutions are obtained uniformly in the speed of light parameter $c \ge 1$. We furthermore prove that solutions to the relativistic Boltzmann equation converge to solutions of the Newtonian Boltzmann equation in the limit as $c\to\infty$ on arbitrary time intervals $[0,T]$, with convergence rate $1/c^{2-\epsilon}$ for any $\epsilon \in(0,2)$. This may be the first proof of unique global in time validity of the Newtonian limit for a Kinetic equation.