Global Solutions to the Ultra-Relativistic Euler Equations

TL;DR

This paper proves a global existence theorem for the ultra-relativistic Euler equations in one dimension, covering a family of equations of state including ideal gases and radiation, using a Nishida-type method and Glimm scheme.

Contribution

It establishes the first global existence results for the ultra-relativistic Euler equations with a broad class of equations of state satisfying thermodynamic laws.

Findings

Global solutions exist for large data in the ultra-relativistic limit.

The analysis applies to equations of state for ideal gases and radiation.

The method combines Nishida-type analysis with Glimm scheme techniques.

Abstract

We prove a global existence theorem for the $3\times 3$ system of relativistic Euler equations in one spacial dimension. It is shown that in the ultra-relativistic limit, there is a family of equations of state that satisfy the second law of thermodynamics for which solutions exist globally. With this limit and equation of state, which includes equations of state for both an ideal gas and one dominated by radiation, the relativistic Euler equations can be analyzed by a Nishida-type method leading to a large data existence theorem, including the entropy and particle number evolution, using a Glimm scheme. Our analysis uses the fact that the equations of state are of the form $p=p(n,S)$, but whose form simplifies to $p=a^{2}\rho$ when viewed as a function of $\rho$ alone.