Hilbert Expansion from the Boltzmann equation to relativistic Fluids

TL;DR

This paper establishes a rigorous connection between the relativistic Boltzmann equation and relativistic Euler equations using a Hilbert expansion, demonstrating that solutions of the former approximate those of the latter in the small Knudsen number regime.

Contribution

It proves the existence of local solutions to the relativistic Boltzmann equation near relativistic Maxwellians derived from solutions to the relativistic Euler equations, including a broad class of near-constant states.

Findings

Solutions to the relativistic Boltzmann equation are close to relativistic Maxwellians.

The dynamics of Boltzmann solutions are effectively governed by relativistic Euler equations in the small Knudsen number limit.

The work covers a large subclass of non-vacuum, near-constant fluid states.

Abstract

We study the local-in-time hydrodynamic limit of the relativistic Boltzmann equation using a Hilbert expansion. More specifically, we prove the existence of local solutions to the relativistic Boltzmann equation that are nearby the local relativistic Maxwellian constructed from a class of solutions to the relativistic Euler equations that includes a large subclass of near-constant, non-vacuum fluid states. In particular, for small Knudsen number, these solutions to the relativistic Boltzmann equation have dynamics that are effectively captured by corresponding solutions to the relativistic Euler equations.