The relativistic Euler equations: ESI notes on their geo-analytic structures and implications for shocks in $1D$ and multi-dimensions

TL;DR

This paper explores the geometric and analytic structures of the relativistic Euler equations, providing detailed constructions of shock formation in 1D and discussing implications for multi-dimensional shocks, with insights into well-posedness and open problems.

Contribution

It offers the first detailed construction of shock formation in 1D using Riemann invariants and discusses new second-order formulations for 3D, linking geometric structures to shock analysis.

Findings

Construction of localized maximal globally hyperbolic developments in 1D

Analysis of the singular boundary and Cauchy horizon in shock solutions

Discussion of geometric structures and implications for multi-dimensional shocks

Abstract

In this article, we provide notes that complement the lectures on the relativistic Euler equations and shocks that were given by the second author at the program Mathematical Perspectives of Gravitation Beyond the Vacuum Regime, which was hosted by the Erwin Schrodinger International Institute for Mathematics and Physics in Vienna in February, 2022. We set the stage by introducing a standard first-order formulation of the relativistic Euler equations and providing a brief overview of local well-posedness in Sobolev spaces. Then, using Riemann invariants, we provide the first detailed construction of a localized subset of the maximal globally hyperbolic developments of an open set of initially smooth, shock-forming isentropic solutions in 1D, with a focus on describing the singular boundary and the Cauchy horizon that emerges from the singularity. Next, we provide an overview of the new second-order formulation of the 3D relativistic Euler equations derived in [41], its rich geometric and analytic structures, their implications for the mathematical theory of shock waves, and their connection to the setup we use in our 1D analysis of shocks. We then highlight some key prior results on the study of shock formation and related problems. Furthermore, we provide an overview of how the formulation of the flow derived in [41] can be used to study shock formation in multiple spatial dimensions. Finally, we discuss various open problems tied to shocks.