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The non-isentropic Einstein-Euler system written in a symmetric hyperbolic form

Uwe Brauer, Lavi Karp

TL;DR

This work develops a symmetric hyperbolic formulation for the non-isentropic relativistic Euler (Einstein–Euler) system by using the pressure as a primary variable and employing a fluid-decomposition strategy. To handle degeneracy near vacuum, it introduces a Makino-type regularization, deriving conditions under which the system remains symmetric and well-posed and providing an explicit Makino variable for common EOS choices. The analysis clarifies the characteristic structure, confirming the flow and sound-cone as the principal characteristics and establishing positive-definiteness of the system's primary matrix under the subluminal sound speed. Together, these results lay the groundwork for local well-posedness theorems for the non-isentropic Einstein–Euler system and provide practical tools for treating vacuum regions in relativistic fluid dynamics.

Abstract

We cast the non--isentropic relativistic Euler system into a symmetric hyperbolic form. Such systems are very suited to treat initial value problems of hyperbolic type. We obtain this form by using the pressure $p$ and not the density $ρ$ as a variable. However, the system becomes degenerate when the pressure $p$ approaches zero, and in these cases we regularise the system by replacing the pressure with an appropriate new matter variable, the Makino variable.

The non-isentropic Einstein-Euler system written in a symmetric hyperbolic form

TL;DR

This work develops a symmetric hyperbolic formulation for the non-isentropic relativistic Euler (Einstein–Euler) system by using the pressure as a primary variable and employing a fluid-decomposition strategy. To handle degeneracy near vacuum, it introduces a Makino-type regularization, deriving conditions under which the system remains symmetric and well-posed and providing an explicit Makino variable for common EOS choices. The analysis clarifies the characteristic structure, confirming the flow and sound-cone as the principal characteristics and establishing positive-definiteness of the system's primary matrix under the subluminal sound speed. Together, these results lay the groundwork for local well-posedness theorems for the non-isentropic Einstein–Euler system and provide practical tools for treating vacuum regions in relativistic fluid dynamics.

Abstract

We cast the non--isentropic relativistic Euler system into a symmetric hyperbolic form. Such systems are very suited to treat initial value problems of hyperbolic type. We obtain this form by using the pressure and not the density as a variable. However, the system becomes degenerate when the pressure approaches zero, and in these cases we regularise the system by replacing the pressure with an appropriate new matter variable, the Makino variable.

Paper Structure

This paper contains 11 sections, 1 theorem, 55 equations.

Table of Contents

  1. Introduction
  2. The relativistic Euler equations with entropy
  3. Energy conditions
  4. Examples of an equation of state for the non--isentropic Euler equations
  5. The non--isentropic equations in symmetric hyperbolic form
  6. Symmetric Hyperbolic Systems
  7. Fluid decomposition
  8. Modification of the fluid decomposed system
  9. Symmetric hyperbolic form
  10. Symmetrization and regularization
  11. The Makino variable for the equation of state (\ref{['eq:sec-02-non-isentropic:3']})

Key Result

Theorem 1

Let $\epsilon$ in (eq:density) be nonnegative density function, the pressure $p$ be defined by (E:pressure) and assume conditions (E:EOSAssumptions). Then the Euler equations (E:Euler)-(E:nandulaw) coupled with the constraint (E:uNormalizedcSquared) can be written as a symmetric hyperbolic system. M

Theorems & Definitions (6)

  • Definition 1: Symmetric hyperbolic system
  • Remark 1
  • Remark 2: The pressure as a matter variable
  • Theorem 1
  • proof
  • Remark 3