On the existence of solutions to the relativistic Euler equations in 2 spacetime dimensions with a vacuum boundary

TL;DR

This paper proves the existence of solutions to the relativistic Euler equations in 2D with vacuum boundaries, showing specific behaviors of sound speed and fluid acceleration near the boundary.

Contribution

It establishes the existence of solutions with vacuum boundaries for a class of relativistic Euler equations, including general equations of state.

Findings

Sound speed approaches zero at vacuum boundary

Fluid acceleration remains finite and bounded near boundary

Results extend to a range of equations of state

Abstract

We prove the existence of a wide class of solutions to the isentropic relativistic Euler equations in 2 spacetime dimensions with an equation of state of the form $p=K\rho^2$ that have a fluid vacuum boundary. Near the fluid vacuum boundary, the sound speed for these solutions are monotonically decreasing, approaching zero where the density vanishes. Moreover, the fluid acceleration is finite and bounded away from zero as the fluid vacuum boundary is approached. The existence results of this article also generalize in a straightforward manner to equations of state of the form $p=K\rho^\frac{\gamma+1}{\gamma}$ with $\gamma > 0$.