Lagrangian formulation and a priori estimates for relativistic fluid flows with vacuum

TL;DR

This paper develops a new Lagrangian symmetrization method for relativistic and non-relativistic compressible fluids with vacuum, providing a priori estimates crucial for understanding free boundary problems in fluid dynamics.

Contribution

It introduces a novel Lagrangian formulation and symmetrization that handle regularity loss near free boundaries in relativistic fluid flows.

Findings

Established a priori estimates in weighted Sobolev spaces

Unified treatment of relativistic and non-relativistic flows

Addressed regularity issues near vacuum boundaries

Abstract

We study the evolution of a compressible fluid surrounded by vacuum and introduce a new symmetrization in Lagrangian coordinates that allows us to encompass both relativistic and non-relativistic fluid flows. The problem under consideration is a free boundary problem of central interest in compressible fluid dynamics and, from the mathematical standpoint, the main challenge to be overcome lies in the loss of regularity in the fluid variables near the free boundary. Based on our Lagrangian formulation, we establish the necessary a priori estimates in weighted Sobolev spaces which are adapted to this loss of regularity.