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

TL;DR

This paper develops a comprehensive theory for self-similar, radially symmetric solutions of relativistic Euler equations incorporating Synge energies, covering classical to ultra-relativistic regimes, and establishes key properties of these solutions.

Contribution

It introduces existence, uniqueness, and structural properties of self-similar solutions with Synge energies across all relativistic regimes, extending classical flow theory.

Findings

Proved existence and uniqueness of solutions for all relativistic parameters.

Established the negativity of the second derivative of Synge energies with respect to pressure.

Demonstrated the monotonic dependence of characteristic velocity on the relativistic parameter.

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 $\gamma = mc^{2}/(k_{B}T)$, thus covering both the classical limit $(\gamma \to \infty)$ and the ultra-relativistic regime $(\gamma \to 0)$. 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 $\gamma$. These results extend the classical theory of self-similar flows to the relativistic framework with kinetic-theory-based constitutive equations.