Concentration in vanishing pressure limit of solutions to the modified Chaplygin gas equations

Two kinds of occurrence mechanism on the phenomenon of concentration and the formation of delta shock wave in vanishing pressure limit of solutions to the modified Chaplygin gas equations are analyzed and identified. The Riemann problem of the modified Chaplygin gas equations is first solved. Then it is shown that, as the pressure vanishes, any two-shock Riemann solution tends to a delta-shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted δ-measure which forms a delta shock wave; any two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution to the transport equations, and the nonvacuum intermediate state in between tends to a vacuum state. It is also shown that, as the pressure approaches the generalized Chaplygin gas pressure, any two-shock Riemann solution tends to a delta-shock solution to the generalized Chaplygin gas equations. Some numerical results are presented to show the formation process of delta shock waves and vacuum states.