Delta-shock waves and Riemann solutions to the generalized pressureless Euler equations with a composite source term

Abstract

In this paper, we are concerned with the Riemann problem for the generalized pressureless Euler equations with a composite source term, which covers the important Eulerian droplet model associated with aircrafts and airways. The Riemann solutions exhibit exactly two types of structures: delta-shocks and vacuum solutions. The generalized Rankine–Hugoniot conditions of the delta shock wave are established and the exact position, propagation speed and strength of the delta shock wave are given explicitly. It is shown that the composite source term makes contact discontinuities and delta shock waves bend into curves and the Riemann solutions are not self-similar anymore, which is a new and interesting phenomenon different from the homogeneous generalized pressureless Euler equations. On the other hand, compared with previous results on the nonhomogeneous generalized pressureless Euler equations, different from the nonhomogeneous case with friction where the state variable u changes linearly with respect to t, here u changes exponentially with respect to t under the influence of composite source term, but with a velocity quite different from the nonhomogeneous case with dissipation. It is also shown that, as the composite source term vanishes wholly or partly, the Riemann solutions converge to the corresponding ones of the homogeneous system or the nonhomogeneous system. These results will give us valuable insights into later research on the (generalized) pressureless Euler equations and other conservation laws with more complicated source terms, such as discontinuous source terms or singular source terms. Finally, two typical examples are given to show the application of our results on the Eulerian droplet model and the nonlinear geometric optics system with a source term.

KEYWORDS:

• Generalized pressureless Euler equations
• Riemann problem
• delta shock wave
• vacuum state
• composite source term

2010 Mathematics Subject Classifications:

• 35L65
• 35L67
• 35B30
• 76N10

Acknowledgments

The authors are grateful to the anonymous referees for his/her valuable comments and corrections, which helped to improve the manuscript. This work is partially supported by Guidance Project of Science Research Program of Hubei Education Department (Grant No. B2019239).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Guidance Project of Science Research Program of Hubei Education Department [B2019239].