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Self-Similar Radially Symmetric Solutions of the Relativistic Euler Equations with Synge Energy

Tommaso Ruggeri, Ferdinand Thein, Qinghua Xiao

TL;DR

The authors study self-similar, radially symmetric solutions of the relativistic Euler equations with Synge-based equations of state for monoatomic and diatomic gases, proving existence and uniqueness of global self-similar entropy solutions across all values of the relativistic parameter $\gamma$. The analysis hinges on a rigorous treatment of the Synge energy, in particular the inequality $e_{pp}<0$, which ensures monotone behavior of the characteristic speeds and enables a detailed blow-up/shock analysis. By reducing to a self-similar ODE system with a crucial denominator $g$ and analyzing cases $s\in(0,1/c]$ and $s>1/c$, the paper characterizes when solutions remain continuous versus when a forward shock forms, including a precise resolution of the shock via Hugoniot conditions and a uniqueness result. The spherical piston problem is treated within the same framework, yielding a unique self-similar piston-driven flow; collectively, the results extend classical self-similar theory to relativistic flows with kinetic-theory-based closures and provide rigorous benchmarks for numerical methods in relativistic gas dynamics.

Abstract

We consider self-similar, radially symmetric solutions of the relativistic Euler equations with constitutive relations from relativistic kinetic theory, based on Synge energies for monatomic and its extension to diatomic gases. For the corresponding initial--boundary value problem, including the spherical piston problem, we prove existence and uniqueness of solutions valid for all values of the relativistic parameter $γ= mc^{2}/(k_{B}T)$, thus covering both the classical limit $(γ\to \infty)$ and the ultra-relativistic regime $(γ\to 0)$. We further establish key structural properties of Synge energies, showing the strict negativity of the second derivative with respect to pressure at constant entropy and the monotone dependence of the characteristic velocity on $γ$. These results extend the classical theory of self-similar flows to the relativistic framework with kinetic-theory-based constitutive equations.

Self-Similar Radially Symmetric Solutions of the Relativistic Euler Equations with Synge Energy

TL;DR

The authors study self-similar, radially symmetric solutions of the relativistic Euler equations with Synge-based equations of state for monoatomic and diatomic gases, proving existence and uniqueness of global self-similar entropy solutions across all values of the relativistic parameter . The analysis hinges on a rigorous treatment of the Synge energy, in particular the inequality , which ensures monotone behavior of the characteristic speeds and enables a detailed blow-up/shock analysis. By reducing to a self-similar ODE system with a crucial denominator and analyzing cases and , the paper characterizes when solutions remain continuous versus when a forward shock forms, including a precise resolution of the shock via Hugoniot conditions and a uniqueness result. The spherical piston problem is treated within the same framework, yielding a unique self-similar piston-driven flow; collectively, the results extend classical self-similar theory to relativistic flows with kinetic-theory-based closures and provide rigorous benchmarks for numerical methods in relativistic gas dynamics.

Abstract

We consider self-similar, radially symmetric solutions of the relativistic Euler equations with constitutive relations from relativistic kinetic theory, based on Synge energies for monatomic and its extension to diatomic gases. For the corresponding initial--boundary value problem, including the spherical piston problem, we prove existence and uniqueness of solutions valid for all values of the relativistic parameter , thus covering both the classical limit and the ultra-relativistic regime . We further establish key structural properties of Synge energies, showing the strict negativity of the second derivative with respect to pressure at constant entropy and the monotone dependence of the characteristic velocity on . These results extend the classical theory of self-similar flows to the relativistic framework with kinetic-theory-based constitutive equations.

Paper Structure

This paper contains 21 sections, 16 theorems, 296 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Governing Equations
  3. Constitutive equations and Synge Energy for monatomic and diatomic gas
  4. Change of variables from $(\rho,T)$ to $(\eta,\gamma)$
  5. Proof of $e_{pp}<0$
  6. Radial symmetry and self-similarity.
  7. Main result and proof for the general radially symmetric case
  8. Investigation of the denominator $g$
  9. The case $s \in (0,1/c]$
  10. The case $s \in (1/c,\infty)$
  11. The case $u_0 \in (0,c)$
  12. The case $u_0 \in (-c,0)$.
  13. The spherical piston problem
  14. Conclusion
  15. Modified Bessel functions and useful estimates
  16. ...and 6 more sections

Key Result

theorem 5.1

Let $(u_0(r),p_0(r),\eta_0(r))\in (-c,c)\times\mathbb{R}_{>0}\times\mathbb{R}_{>0}$ be an initial datum such that and considering the following boundary condition: Then the system Euler31 admits a unique self-similar entropy solution.

Figures (2)

  • Figure 1: Visualization of the situation described in Lem. \ref{['lem:blowup']} and Lem. \ref{['lem:unique_shock']}. For an improved presentation we present the dimensionless quantities which are obtained with a proper scaling by the speed of light $c$. The blue curve represents the solution $u(s)/c$ of the ODE system up to $c\,\bar{s}$. The red curve shows the function $\varphi_A(s)/c$ defined in \ref{['def:phi_A']}. Starting from $c\,\hat{s}$, we compute the perturbed state corresponding to each given unperturbed state defined by the ODE solution at given value of $c\,s$. The quantity $u_\delta(s)/c$ is depicted by the yellow curve. One verifies that all curves coincide at $c\,\bar{s}$ and that $u_\delta(s)/c$ is monotonically decreasing with a unique zero at $c\,s^\ast$ where the shock occurs.
  • Figure 2: Shock solutions for the monoatomic (blue) and diatomic (red) gases with $u_0/c = -1/\sqrt{2}$, $\gamma_0 = 3$ (solid) and $\gamma_0 = 30$ (dashed). In the figure, the $x$–axis is expressed in terms of the dimensionless variable $cs$.

Theorems & Definitions (28)

  • theorem 5.1
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • ...and 18 more