The relativistic Euler equations: Remarkable null structures and regularity properties

TL;DR

This paper introduces a new formulation of the relativistic Euler equations that reveals null structures, enabling better regularity results and aiding the analysis of shock formation in relativistic fluids.

Contribution

The paper presents a novel coupled geometric wave, transport, and elliptic system formulation of the relativistic Euler equations, highlighting null structures and improved regularity properties.

Findings

Established local well-posedness with enhanced regularity for vorticity and entropy.

Demonstrated the formulation's suitability for studying stable shock formation.

Applicable to arbitrary equations of state, not limited to barotropic fluids.

Abstract

We derive a new formulation of the relativistic Euler equations that exhibits remarkable properties. This new formulation consists of a coupled system of geometric wave, transport, and elliptic equations, sourced by nonlinearities that are null forms relative to the acoustical metric. Our new formulation is well-suited for various applications, in particular for the study of stable shock formation, as it is surveyed in the paper. Moreover, using the new formulation presented here, we establish a local well-posedness result showing that the vorticity and the entropy of the fluid are one degree more differentiable compared to the regularity guaranteed by standard estimates (assuming that the initial data enjoy the extra differentiability). This gain in regularity is essential for the study of shock formation without symmetry assumptions. Our results hold for an arbitrary equation of state, not necessarily of barotropic type.