The Riemann problem and wave interactions for a relativistic $ p $-system with variable heat ratios

Jiatao Yao; Qinglong Zhang; Jiatao Yao; Qinglong Zhang

doi:10.3934/math.2026680

AIMS Mathematics

2026, Volume 11, Issue 6: 16570-16601. doi: 10.3934/math.2026680

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Research article

The Riemann problem and wave interactions for a relativistic $ p $-system with variable heat ratios

Jiatao Yao ,
Qinglong Zhang ^,

School of Mathematics and Statistics, Ningbo University, 315211, China

Received: 09 March 2026 Revised: 20 May 2026 Accepted: 28 May 2026 Published: 09 June 2026
MSC : 35L65, 35L80, 35R35

We comprehensively investigate the Riemann problem for a relativistic $ p $-system with variable heat ratios. First, we consider the relativistic system when the heat ratio $ \gamma $ changes. We supplement with the trivial equation $ \frac{\partial}{\partial t}\gamma = 0 $ to close the system. Second, we solve the Riemann problem with piecewise constant of heat ratios, and obtain the elementary waves, including rarefaction waves, shock waves, contact discontinuities, and particularly stationary waves. An admissible stationary wave selection method based on monotone criterion is studied. Finally, we discuss the interaction of elementary waves, particularly, rarefaction wave and shock wave interactions with stationary wave, respectively. The large-time behavior of each case is also presented.
- the Riemann problem,
- relativistic system,
- variable heat ratios,
- stationary wave,
- wave interactions
Citation: Jiatao Yao, Qinglong Zhang. The Riemann problem and wave interactions for a relativistic $ p $-system with variable heat ratios[J]. AIMS Mathematics, 2026, 11(6): 16570-16601. doi: 10.3934/math.2026680

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Abstract

We comprehensively investigate the Riemann problem for a relativistic $ p $-system with variable heat ratios. First, we consider the relativistic system when the heat ratio $ \gamma $ changes. We supplement with the trivial equation $ \frac{\partial}{\partial t}\gamma = 0 $ to close the system. Second, we solve the Riemann problem with piecewise constant of heat ratios, and obtain the elementary waves, including rarefaction waves, shock waves, contact discontinuities, and particularly stationary waves. An admissible stationary wave selection method based on monotone criterion is studied. Finally, we discuss the interaction of elementary waves, particularly, rarefaction wave and shock wave interactions with stationary wave, respectively. The large-time behavior of each case is also presented.

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