On a Class of Global Solutions to 3D Free-Boundary Relativistic Euler Equations with a Physical Vacuum Boundary

TL;DR

This paper constructs a class of spherically symmetric, future-global solutions to 3D relativistic Euler equations with a physical vacuum boundary, allowing for near-light speed expansion, and extends understanding beyond classical Euler models.

Contribution

It introduces a new family of global solutions for relativistic Euler equations with physical vacuum, not just perturbations of classical models.

Findings

Solutions expand asymptotically linearly in time

Expansion rate can approach the speed of light

Solutions are spherically symmetric with small initial density

Abstract

We consider the free-boundary relativistic Euler equations in Minkowski spacetime $\mathbb{M}^{1+3}$ equipped with a physical vacuum boundary, which models the motion of a relativistic gas. We concern ourselves with the family of isentropic, barotropic, and polytropic gas, with an equation of state $p = \rho^{1+\kappa}, \kappa \in (0,\frac23]$. We construct an open class of initial data that launches future-global solutions. Such solutions are spherically symmetric, have small initial density, and expand asymptotically linearly in time. In particular, the asymptotic rate of expansion is allowed to be arbitrarily close to the speed of light. Therefore, our main result is far from a perturbation of existing results concerning the classical isentropic Euler counterparts.