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

TL;DR

This paper investigates the Riemann problem for pressureless Euler equations with a discontinuous source term, deriving delta shock wave solutions and observing phenomena like shock disappearance and vacuum formation.

Contribution

It introduces a method combining generalized Rankine-Hugoniot conditions with characteristics to solve the problem with discontinuous sources, revealing new shock behaviors.

Findings

Delta shock wave solutions are obtained for various cases.

Disappearance of delta shocks and vacuum states are observed.

The source term significantly influences shock dynamics.

Abstract

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 different situations, which reflects the impact of the source term on the delta shock front. Moreover, during the construction process of 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.