The Wave Interactions for the Chaplygin Gas Dynamics With a Dirac Measure Source Term

ABSTRACT In this paper, we concern with wave interaction for the Chaplygin gas dynamics with a Dirac measure source term, in which the delta standing wave, delta shock wave, contact discontinuity and stationary contact discontinuity are involved. A general framework is provided to investigate the wave interaction problem. Under this framework, we analyze the possible wave patterns in each case, and obtain all of exact solutions which rigorously show the evolution process of various waves. Intriguingly, it is observed that when a contact discontinuity collides with a stationary contact discontinuity, it then passes through the stationary contact discontinuity and evolves a delta shock wave. It is also noticed that when a delta shock wave interacts with a stationary contact discontinuity, it then penetrates the stationary contact discontinuity and bifurcates multiple delta contact discontinuities. What is more, after the interaction between a contact discontinuity and a delta standing wave, they evolve either a delta shock wave followed by a stationary contact discontinuity, or multiple delta contact discontinuities followed by a stationary contact discontinuity. Finally, our technique and results can contribute to the studies on wave interactions for quasilinear hyperbolic systems with a general singular source term.