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This article discusses the Riemann problem for the Chaplygin gas Euler equations that include the presence of two source terms. By means of variable substitution, two kinds of non-self-similar Riemann solutions involving delta-shock are constructed explicitly. For the delta-shock, the generalized Rankine–Hugoniot relations and the over-compressive entropy condition are clarified. Moreover, the position, propagation speed and strength of the delta-shock are given explicitly. It is discovered that the position of the delta-shock is a combination of an exponential function and a linear function, and the weight of the delta-shock is an exponential function of the time. Interestingly, even when the delta-shock is a straight line, the weight of the delta-shock is no longer a linear function of the time  t . In addition, it is proved that the Riemann solutions converge to the corresponding ones of Chaplygin gas Euler equations with friction as  k  drops to zero, and the Riemann solutions converge to the corresponding ones of Chaplygin gas Euler equations as  k  and  β  tend to zero simultaneously. Furthermore, it is also shown that the limits of Riemann solutions are just the Riemann solutions to the transport equations with same source terms as the Chaplygin gas pressure falls to zero.

Delta-shock for the Chaplygin gas Euler equations with source terms