Concentration and cavitation phenomena of Riemann solutions for the generalized Chaplygin gas equations under the flux approximation

Abstract

In this paper, we investigate the concentration and cavitation phenomena of Riemann solutions for the generalized Chaplygin gas equations in the presence of flux approximation. The concentration and cavitation are fundamental and physical phenomena in fluid dynamics, which can be mathematically described by delta shock waves and vacuums (or constant density states), respectively. The main objective of this paper is to rigorously investigate the formation of delta shock waves and constant density states and observe the concentration and cavitation phenomena. First, the Riemann problem for the generalized Chaplygin gas equations under the flux approximation is solved constructively. 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 1-shock wave curve and the 2-shock wave curve do not intersect each other in the phase plane. Second, it is rigorously proved that, as the pressure vanishes, the Riemann solutions for the generalized Chaplygin gas equations under the flux approximation tend to the two kinds of Riemann solutions to the transport equations in zero-pressure flow under the flux approximation, which include a delta shock wave formed by a weighted δ-measure and a constant density state.