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On a Class of Global Solutions to 3D Free-Boundary Relativistic Euler Equations with a Physical Vacuum Boundary

Marcelo M. Disconzi, Zhongtian Hu, Chenyun Luo

TL;DR

The article constructs a robust global-in-time framework for free-boundary relativistic Euler flows with a physical vacuum in Minkowski space, focusing on isentropic polytropic gases with $p(\rho)=\rho^{1+\kappa}$ and $\kappa\in(0,2/3]$. By imposing a mass-critical scaling and working in a spherically symmetric, Lagrangian setting, the authors derive a perturbed system around a linearly expanding background, introduce a relativistic correction $\mathcal{G}$ and a good unknown $\Theta$, and develop a weighted energy method with a careful elliptic-pressure decomposition. They establish a hierarchy of high-order energies $\mathcal{E}^N$ and a main energy inequality that closes via a bootstrap argument, yielding future-global solutions with asymptotically linear expansion and subluminal sound speed. The analysis reveals delicate cancellations and a lack of scaling symmetry in the relativistic system, which the authors overcome through structural decompositions and a mass-critical framework, providing a rigorous expansion-stability theory for relativistic gases with a physical vacuum boundary. This work advances understanding of long-time dynamics for relativistic fluids and offers a template for studying stability of expanding astrophysical gases in general relativistic settings. $\,$

Abstract

We consider the free-boundary relativistic Euler equations in Minkowski spacetime $\mathbb{M}^{1+3}$ equipped with a physical vacuum boundary, which models the motion of a relativistic gas. We concern ourselves with the family of isentropic, barotropic, and polytropic gas, with an equation of state $p = ρ^{1+κ}, κ\in (0,\frac23]$. We construct an open class of initial data that launches future-global solutions. Such solutions are spherically symmetric, have small initial density, and expand asymptotically linearly in time. In particular, the asymptotic rate of expansion is allowed to be arbitrarily close to the speed of light. Therefore, our main result is far from a perturbation of existing results concerning the classical isentropic Euler counterparts.

On a Class of Global Solutions to 3D Free-Boundary Relativistic Euler Equations with a Physical Vacuum Boundary

TL;DR

The article constructs a robust global-in-time framework for free-boundary relativistic Euler flows with a physical vacuum in Minkowski space, focusing on isentropic polytropic gases with and . By imposing a mass-critical scaling and working in a spherically symmetric, Lagrangian setting, the authors derive a perturbed system around a linearly expanding background, introduce a relativistic correction and a good unknown , and develop a weighted energy method with a careful elliptic-pressure decomposition. They establish a hierarchy of high-order energies and a main energy inequality that closes via a bootstrap argument, yielding future-global solutions with asymptotically linear expansion and subluminal sound speed. The analysis reveals delicate cancellations and a lack of scaling symmetry in the relativistic system, which the authors overcome through structural decompositions and a mass-critical framework, providing a rigorous expansion-stability theory for relativistic gases with a physical vacuum boundary. This work advances understanding of long-time dynamics for relativistic fluids and offers a template for studying stability of expanding astrophysical gases in general relativistic settings.

Abstract

We consider the free-boundary relativistic Euler equations in Minkowski spacetime equipped with a physical vacuum boundary, which models the motion of a relativistic gas. We concern ourselves with the family of isentropic, barotropic, and polytropic gas, with an equation of state . We construct an open class of initial data that launches future-global solutions. Such solutions are spherically symmetric, have small initial density, and expand asymptotically linearly in time. In particular, the asymptotic rate of expansion is allowed to be arbitrarily close to the speed of light. Therefore, our main result is far from a perturbation of existing results concerning the classical isentropic Euler counterparts.
Paper Structure (78 sections, 54 theorems, 519 equations)

This paper contains 78 sections, 54 theorems, 519 equations.

Table of Contents

  1. Introduction
  2. History and Background
  3. Main Difficulties and Methodology
  4. Methodology
  5. Organization of the Paper
  6. List of Notations
  7. A Lagrangian Formulation of relativistic Euler Equations in Spherical Symmetry
  8. Relativistic Euler Equations in Spherical Symmetry
  9. Scaling Ansatz
  10. Modified Velocity
  11. Lagrangian Formulation
  12. Momentum Equation in Lagrangian Coordinates
  13. Mass Equation in Lagrangian Coordinates
  14. Admissible Weight and Enthalpy Profile
  15. Equation for Perturbation Variables
  16. ...and 63 more sections

Key Result

Theorem 1.1

(Main theorem: a Rough Version) Given $\kappa \in (0,\frac{2}{3}]$, $(\lambda_0,\lambda_1) \in \mathbb{R}^+ \times (0,1)$, there exists an open set of spherically symmetric initial data supported in $B(0,\lambda_0)$ with initial expanding rate $\lambda_1$ and initial mass $\delta^\frac{1}{\kappa} \l

Theorems & Definitions (112)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • ...and 102 more