A priori estimates for solutions to the relativistic Euler equations with a moving vacuum boundary

TL;DR

This paper derives local-in-time a priori estimates for solutions to the relativistic Euler equations with a moving vacuum boundary, addressing the degeneracy at the fluid-vacuum interface in a relativistic setting.

Contribution

It extends the analysis of vacuum boundary problems to the relativistic Euler equations, providing estimates in Lagrangian coordinates where standard methods fail.

Findings

Established local-in-time a priori estimates for relativistic Euler solutions.

Analyzed the degenerate hyperbolic system near the vacuum boundary.

Identified the relativistic analogs of physical vacuum conditions.

Abstract

We study the relativistic Euler equations on the Minkowski spacetime background. We make assumptions on the equation of state and the initial data that are relativistic analogs of the well-known physical vacuum boundary condition, which has played an important role in prior work on the non-relativistic compressible Euler equations. Our main result is the derivation, relative to Lagrangian (also known as co-moving) coordinates, of local-in-time a priori estimates for the solution. The solution features a fluid-vacuum boundary, transported by the fluid four-velocity, along which the hyperbolicity of the equations degenerates. In this context, the relativistic Euler equations are equivalent to a degenerate quasilinear hyperbolic wave-map-like system that cannot be treated using standard energy methods.