A priori estimates for relativistic liquid bodies

TL;DR

This paper reformulates the relativistic Euler equations for liquid bodies into a wave equation system with acoustic boundary conditions, establishing energy estimates without loss of derivatives and without requiring irrotationality.

Contribution

It introduces a new wave formulation for relativistic liquid bodies that avoids divergence and curl estimates, broadening the scope of energy estimate techniques.

Findings

Energy estimates without loss of derivatives are proved.

The wave formulation applies to non-irrotational flows.

The approach simplifies analysis of relativistic liquid bodies.

Abstract

We demonstrate that a sufficiently smooth solution of the relativistic Euler equations that represents a dynamical compact liquid body, when expressed in Lagrangian coordinates, determines a solution to a system of non-linear wave equations with acoustic boundary conditions. Using this wave formulation, we prove that these solutions satisfy energy estimates without loss of derivatives. Importantly, our wave formulation does not require the liquid to be irrotational, and the energy estimates do not rely on divergence and curl type estimates employed in previous works.