A Priori Estimates for the Linearized Relativistic Euler Equations with a Physical Vacuum Boundary and an Ideal Gas Equation of State

Abstract

In this work, we will provide a key result on the relativistic Euler equations with an ideal gas equation of state and a physical vacuum boundary. More specifically, we will prove a priori estimates for the linearized system in weighted Sobolev spaces. As seen in works such as (Disconzi et al., 2022), analysis of the linearized equations often plays a crucial role in developing a local well-posedness theory for the non-linear problem, and our main result is an important step in this direction for the ideal gas equation of state. Our focus will be on choosing the correct thermodynamic variables, developing a weighted book-keeping scheme, and then providing energy estimates for the linearized system.