The limits of Riemann solutions to the relativistic van der Waals fluid

Abstract

By a Riemann problem, we identify and analyze the phenomena of concentration and cavitation and the formation of delta-shock waves and vacuum states in solutions to the relativistic van der Waals fluid as the pressure and molecular excluded volume drop to zero simultaneously. Particularly, the geometric properties of elementary wave curves and the formation process of delta shocks and vacuums, influenced by the parameter of pressure perturbation ε and the van der Waals excluded volume b, are characterized and clarified completely. It is rigorously shown that the Riemann solution of the relativistic van der Waals fluid containing two shocks and two-rarefaction waves respectively tend to the delta-shock solution and vacuum state to the pressureless relativistic Euler equations as both pressure and molecular excluded volume vanish.

COMMUNICATED BY:

• M. Mei

Keywords:

• Relativistic Euler equations
• van der Waals gas
• Riemann problem
• delta-shock wave
• vanishing pressure limit

AMS CLASSIFICATIONS:

• 35L65
• 35L67
• 76N10
• 76N15

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is partially supported by Yunnan Applied Basic Research Projects [grant number 2018FD015]; Scientific Research Foundation Project of Yunnan Education Department [grant number 2018JS150]; Science and Technology Foundation of Hebei Education Department [grant number QN2018307]; Natural Science Foundation of Hebei Province [grant number A2019105110].