Uniqueness of self-similar entropy solutions for the radially symmetric relativistic Euler equations with a polytropic equation of state

For the radially symmetric relativistic Euler equations in two and three space dimensions, Lai (2020) constructed self-similar entropy solutions to the Riemann initial-boundary value problem for the ultra-relativistic equation of state $p=\rho/3$ and the polytropic (gamma-law) equation of state $\rho = n + p/(\gamma-1)$ , $p = S n^\gamma$ with $1<\gamma<5/3$ , but could prove uniqueness only in the ultra-relativistic case. Whether the self-similar entropy solution — in particular the shock location for incoming initial data — is unique in the full-relativistic polytropic case remains without a stated theorem. A recent non-peer-reviewed preprint (Ruggeri–Thein–Xiao, 2025) proves existence and uniqueness for the physically distinct Synge-energy equation of state, including the spherical piston problem, and asserts polytropic coverage only through the classical low-temperature limit. The problem asks for a proof or disproof of uniqueness of the self-similar entropy solution for the polytropic equation of state in the genuinely relativistic regime, for both the Riemann initial-boundary value problem and the self-similar spherical piston problem.

Background

The relativistic Euler equations govern the motion of a compressible gas in special relativity, expressing conservation of particle number and of energy–momentum through the particle four-flux and the energy–momentum tensor of a perfect fluid, with all flow speeds bounded by the light speed $c$ . Closing the system requires an equation of state relating the pressure $p$ , the particle density $n$ , and the energy density $\rho$ . Two classical choices are the ultra-relativistic law $p=\rho/3$ , appropriate when thermal energy dominates rest-mass energy, and the polytropic (gamma-law) form $\rho = n + p/(\gamma-1)$ with $p = S n^\gamma$ and adiabatic exponent $1<\gamma<5/3$ , which models a relativistic ideal gas with rest mass; the bound $\gamma<5/3$ keeps the sound speed subluminal. A weaker solution is called an entropy solution if it satisfies the entropy inequality across discontinuities, so that shocks are physical rather than rarefaction shocks. Because multidimensional Riemann problems for these systems are beyond current techniques, one studies radially symmetric reductions, which model explosion waves (a constant ambient state driven by data at the center, with a vanishing-flux symmetry condition at the origin) and the spherical piston problem (a sphere $r = R(t)$ expanding into a medium at a prescribed radial speed, driving a shock). In radial symmetry one seeks self-similar solutions depending only on $s = t/r$ (equivalently the similarity variable $\xi = r/t$ ); the PDE system reduces to a non-autonomous system of nonlinear ordinary differential equations coupled, when shocks are present, to the relativistic Rankine–Hugoniot jump conditions (the conservation-law relations that connect the flow states on the two sides of a shock to the shock's speed). For initial velocity $u_0<0$ (inward motion), the solution contains a single shock placed at some self-similar location $s_s$ , and the construction requires solving for the post-shock velocity $u_2(s)$ as a function of a candidate location $s$ . Lai (2020) proved existence of self-similar entropy solutions for both equations of state for the Riemann initial-boundary value problem, but proved uniqueness only for the ultra-relativistic law; his Remark 3.1 records that uniqueness of the shock location $s_s$ for $u_0<0$ in the relativistic case was open because the needed monotone decrease of $u_2(s)$ in $s$ was difficult to establish. His paper also contains the comparison-principle material (his Lemma 2.6) that the later piston analysis of Ruggeri–Thein–Xiao invokes. For context, the one-dimensional relativistic Riemann problem and the non-relativistic radially symmetric theory were previously settled, and subsequent work constructed related radial solutions (piecewise smooth single-shock solutions with a priori Riemann-invariant estimates by Hu–Zhang, piston solutions for $p=\sigma^2\rho$ by Lu–Kao–Lien, global smooth shock-free solutions by Fan–Lai–Yuan, and ultra-relativistic benchmarks by Kunik and collaborators), none of which addresses the polytropic uniqueness question. Most recently, Ruggeri–Thein–Xiao (non-peer-reviewed preprint) proved existence and uniqueness of self-similar solutions for the Riemann initial-boundary value problem and the piston problem for the Synge-energy equation of state (a constitutive law from relativistic kinetic theory involving modified Bessel functions, valid at all relativistic temperatures), with the key step being precisely a monotonicity/uniqueness argument for the shock location; they state in a remark that this settles the question noted in Lai's Remark 3.1 for the polytropic case, but that assertion is derived from the classical low-temperature limit of the Synge family (in which its relativistic temperature parameter $mc^2/(k_B T)$ tends to infinity) and no theorem is stated for the polytropic family $p = S n^\gamma$ , $1<\gamma<5/3$ , at genuinely relativistic temperatures.

Problem Statement

For the relativistic Euler equations in two or three space dimensions under radial symmetry, with the polytropic equation of state $\rho = n + p/(\gamma-1)$ , $p = S n^\gamma$ with $1<\gamma<5/3$ (the full-relativistic case of Lai's construction), settle the uniqueness of the self-similar entropy solutions: (i) for the Riemann initial-boundary value problem with constant initial data and initial velocity $u_0<0$ , whose existence is known from Lai (2020), prove that the self-similar entropy solution is unique — equivalently, that the shock location $s_s$ in the self-similar variable $s=t/r$ is unique — or exhibit non-uniqueness; and (ii) for the self-similar spherical piston problem, whose existence is known from Lai (2020), prove that the self-similar entropy solution is unique — equivalently, that the shock location is unique — or exhibit non-uniqueness (Ruggeri–Thein–Xiao establish existence and uniqueness for the Synge-energy family in their Theorem 6.1, with a proof invoking Lai's Lemma 2.6 comparison argument; no verified published theorem establishes uniqueness for the polytropic piston case). Both statements are to be established for the polytropic relativistic system itself, not inferred solely from the classical low-temperature limit of the Synge-energy family. A rigorous counterexample exhibiting two distinct self-similar entropy solutions of the same problem equally resolves the question.

References

Geng Lai, "Self-similar solutions of the radially symmetric relativistic Euler equations", European Journal of Applied Mathematics 31(6), 919–949 (2020). DOI: 10.1017/S0956792519000317, https://doi.org/10.1017/S0956792519000317
Tommaso Ruggeri, Ferdinand Thein, Qinghua Xiao, "Self-Similar Radially Symmetric Solutions of the Relativistic Euler Equations with Synge Energy", preprint (2025). arXiv:2511.18971, https://arxiv.org/abs/2511.18971
Geng Lai, "Global solution to a three-dimensional spherical piston problem for the relativistic Euler equations", European Journal of Applied Mathematics 33(1), 1–26 (published online September 2020). DOI: 10.1017/S0956792520000315, https://doi.org/10.1017/S0956792520000315
Yanbo Hu, Binyu Zhang, "On the analytical construction of radially symmetric solutions for the relativistic Euler equations", Journal of the London Mathematical Society 110(5) (2024). DOI: 10.1112/jlms.70005, https://doi.org/10.1112/jlms.70005
Bing-Ze Lu, Chou Kao, Wen-Ching Lien, "Self-similar solutions of the relativistic Euler system with spherical symmetry", Quarterly of Applied Mathematics 82(4), 705–734 (2024). DOI: 10.1090/qam/1680, https://doi.org/10.1090/qam/1680
Junlong Fan, Geng Lai, Zijun Yuan, "Global existence of smooth solutions to spherically symmetric relativistic Euler equations", Journal of Mathematical Physics 66(12) (2025). DOI: 10.1063/5.0264180, https://doi.org/10.1063/5.0264180
Geng Lai, Zijun Yuan, Yingchun Shi, "On the expansion of a flow into vacuum for spherically symmetric relativistic hydrodynamic equations", Journal of Mathematical Analysis and Applications 543, article 128922 (2025). DOI: 10.1016/j.jmaa.2024.128922, https://doi.org/10.1016/j.jmaa.2024.128922
Matthias Kunik, Adrian Kolb, Siegfried Müller, Ferdinand Thein, "Radially symmetric solutions of the ultra-relativistic Euler equations in several space dimensions", Journal of Computational Physics 518, article 113330 (2024). arXiv:2402.12857, DOI: 10.1016/j.jcp.2024.113330, https://arxiv.org/abs/2402.12857
Matthias Kunik, Hailiang Liu, Gerald Warnecke, "Radially symmetric solutions of the ultra-relativistic Euler equations", Methods and Applications of Analysis 28(4), 401–422 (2021). arXiv:2002.01181, DOI: 10.4310/maa.2021.v28.n4.a1, https://arxiv.org/abs/2002.01181
Ferdinand Thein, Hendrik Ranocha, "Computing Radially-Symmetric Solutions of the Ultra-Relativistic Euler Equations with Entropy-Stable Discontinuous Galerkin Methods", Journal of Computational Physics 561, article 114959 (2026). arXiv:2508.21427, DOI: 10.1016/j.jcp.2026.114959, https://arxiv.org/abs/2508.21427