Zhiqiang Shao
In this paper, we investigate the limiting behavior of Riemann solutions to the generalized Chaplygin gas equations with damping and friction when the pressure vanishes. Unlike the homogeneous case, the Riemann solutions are no longer self-similar. First, we use a transformation to solve the Riemann problem for the generalized Chaplygin gas equations with damping and friction. Although the system is strictly hyperbolic and its two characteristic fields are genuinely nonlinear, the delta shock wave arises in Riemann solutions. The formation of mechanism for delta shock wave is analyzed; that is, the one-shock wave curve and the two-shock wave curve do not intersect each other in the phase plane. Second, it is rigorously proved that, when the pressure vanishes, a Riemann solution consisting of two shock waves converges to a delta shock wave solution of the transport equations in zero-pressure flow, while a Riemann solution consisting of two rarefaction waves converges to a solution consisting of four contact discontinuities together with vacuum states with three different virtual velocities in the limiting scenario.