Recent developments in mathematical aspects of relativistic fluids

TL;DR

This paper reviews recent mathematical advances in relativistic fluid dynamics, focusing on Euler equations, shock formation, low-regularity solutions, vacuum boundaries, and viscosity, aiming to guide researchers and students.

Contribution

It provides an accessible overview of recent mathematical developments in relativistic fluids, highlighting new formulations, solution behaviors, and open problems.

Findings

Introduction of a new wave-transport formulation of relativistic Euler equations

Analysis of shock formation in relativistic fluids

Discussion of low-regularity solutions and vacuum boundary problems

Abstract

We review some recent developments in mathematical aspects of relativistic fluids. The goal is to provide a quick entry point to some research topics of current interest that is accessible to graduate students and researchers from adjacent fields, as well as to researches working on broader aspects of relativistic fluid dynamics interested in its mathematical formalism. Instead of complete proofs, which can be found in the published literature, here we focus on the proofs' main ideas and key concepts. After an introduction to the relativistic Euler equations, we cover the following topics: a new wave-transport formulation of the relativistic Euler equations tailored to applications; the problem of shock formation for relativistic Euler; rough (i.e., low-regularity) solutions to the relativistic Euler equations; the relativistic Euler equations with a physical vacuum boundary; relativistic fluids with viscosity. We finish with a discussion of open problems and future directions of research.