Dynamical relativistic liquid bodies I: constraint propagation

TL;DR

This paper introduces a new wave formulation for relativistic Euler equations with vacuum boundaries, proving constraint propagation and paving the way for establishing local existence and uniqueness of solutions for relativistic liquid bodies.

Contribution

It presents a novel wave formulation with boundary conditions and proves the propagation of constraints, advancing the mathematical understanding of relativistic fluid dynamics.

Findings

New wave formulation for relativistic Euler equations with vacuum boundary conditions

Proof of constraint propagation in the new formulation

Foundation for local-in-time existence and uniqueness results

Abstract

We introduce a new wave formulation for the relativistic Euler equations with vacuum boundary conditions that consists of a system of non-linear wave equations in divergence form with a combination of acoustic and Dirichlet boundary conditions. We show that solutions of our new wave formulation determine solutions of the relativistic Euler equations that satisfy the vacuum boundary conditions provided the initial data is chosen to make a specific set of constraints vanish on the initial hypersurface. Moreover, we prove that these constraints propagate. This article is the first step of a two step strategy to establish the local-in-time existence and uniqueness of solutions to the relativistic Euler equations representing dynamical liquid bodies in vacuum.