Singular solutions to the Riemann problem for the pressureless Euler equations with discontinuous source term

ABSTRACT

It is interesting and challenging to study conservation laws with discontinuous source terms and explore how the delta shock wave is influenced by the discontinuous source term. However, so far, few results have been obtained about it. In this paper, the Riemann problem for the pressureless Euler equations with a discontinuous source term is considered. The delta shock wave solution is obtained by combining the generalized Rankine-Hugoniot conditions, together with the method of characteristics for various kinds of different situations, and the impact of the discontinuous source term on the delta shock front are precisely illustrated. Moreover, during the process of constructing the Riemann solution, some interesting phenomena are also observed, such as the disappearance of the delta shock wave and the occurrence of the vacuum state, etc.

KEYWORDS:

• Delta shock wave
• vacuum state
• Riemann problem
• pressureless Euler equations
• discontinuous source term

2010 MSC:

• 35L65
• 35L67
• 35B25
• 90B20

Acknowledgments

The authors are grateful to the anonymous referees for his/her valuable comments and corrections, which help to improve the manuscript.

Disclosure statement

The authors declare that they have no conflicts of interest to this work.

Additional information

Funding

This work is supported by University Level Scientific Research Project of Jianghan University [grant number 2021yb059].