Formation mechanism of concentration and entropy-related cavitation in the pressureless limit for non-isentropic logarithmic gas dynamics

Yiming Gao; Weifeng Jiang; Chengkai Tu; Yiming Gao; Weifeng Jiang; Chengkai Tu

doi:10.3934/math.2026969

AIMS Mathematics

2026, Volume 11, Issue 8: 24033-24059. doi: 10.3934/math.2026969

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Formation mechanism of concentration and entropy-related cavitation in the pressureless limit for non-isentropic logarithmic gas dynamics

College of Science, China Jiliang University, Hangzhou 310018, China

Received: 09 June 2026 Revised: 26 July 2026 Accepted: 31 July 2026 Published: 05 August 2026
MSC : 35L65, 35L80, 35R35

In this paper, we study the pressureless limit of the Riemann solutions for a reduced two-phase mixtures model with non-isentropic logarithmic gas state. We construct the Riemann solutions using the $ p-u $ plane projection method and study the formation of delta shock waves and vacuum as the pressure vanishes. We prove that, as pressure vanishes, the limit of Riemann solutions is the Riemann solutions of the reduced 2-dimensional pressureless gas dynamics model, where the evacuation is strictly determined by the state with higher entropic stiffness. In particular, we compare the rates at which the density of the intermediate state tends to singularity, such as infinity or vacuum, when the model is with logarithmic gas, polytropic gas, and generalized Chaplygin gas as the pressure vanishes. Then, we quantitatively demonstrated that logarithmic gases can be regarded as a transitional form between polytropic gases and generalized Chaplygin gases. Finally, numerical simulations employing essentially non-oscillatory schemes with third-order Runge-Kutta methods are used to validate the theoretical convergence behaviors.
- concentration and cavitation,
- Riemann solution,
- vanishing pressure,
- non-isentropic logarithmic gas,
- two-phase mixtures model
Citation: Yiming Gao, Weifeng Jiang, Chengkai Tu. Formation mechanism of concentration and entropy-related cavitation in the pressureless limit for non-isentropic logarithmic gas dynamics[J]. AIMS Mathematics, 2026, 11(8): 24033-24059. doi: 10.3934/math.2026969

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Abstract

In this paper, we study the pressureless limit of the Riemann solutions for a reduced two-phase mixtures model with non-isentropic logarithmic gas state. We construct the Riemann solutions using the $ p-u $ plane projection method and study the formation of delta shock waves and vacuum as the pressure vanishes. We prove that, as pressure vanishes, the limit of Riemann solutions is the Riemann solutions of the reduced 2-dimensional pressureless gas dynamics model, where the evacuation is strictly determined by the state with higher entropic stiffness. In particular, we compare the rates at which the density of the intermediate state tends to singularity, such as infinity or vacuum, when the model is with logarithmic gas, polytropic gas, and generalized Chaplygin gas as the pressure vanishes. Then, we quantitatively demonstrated that logarithmic gases can be regarded as a transitional form between polytropic gases and generalized Chaplygin gases. Finally, numerical simulations employing essentially non-oscillatory schemes with third-order Runge-Kutta methods are used to validate the theoretical convergence behaviors.

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