Hydrodynamic limit and Newtonian limit from the relativistic Boltzmann equation to the classical Euler equations

TL;DR

This paper rigorously justifies the hydrodynamic and Newtonian limits of the relativistic Boltzmann equation to classical Euler equations, providing convergence rates without assuming dependence between Knudsen number and light speed.

Contribution

It establishes the validity of two independent limits from relativistic to classical kinetic theory without prior dependence assumptions, using Hilbert expansion and uniform estimates.

Findings

Proved convergence of relativistic Boltzmann to Euler equations.

Derived explicit convergence rates.

Overcame difficulties in uniform estimates for relativistic operators.

Abstract

The hydrodynamic limit and Newtonian limit are important in the relativistic kinetic theory. We justify rigorously the validity of the two independent limits from the special relativistic Boltzmann equation to the classical Euler equations without assuming any dependence between the Knudsen number $\varepsilon$ and the light speed $\mathfrak{c}$. The convergence rates are also obtained. This is achieved by Hilbert expansion of relativistic Boltzmann equation. New difficulties arise when tacking the uniform in $\mathfrak{c}$ and $\varepsilon$ estimates for the Hilbert expansion, which have been overcome by establishing some uniform-in-$\mathfrak{c}$ estimate for relativistic Boltzmann operators.