FORMATION OF DELTA-SHOCKS FOR THE CHAPLYGIN GAS EQUATIONS BY LOGARITHMIC PRESSURE PERTURBATION

Abstract

We introduce a logarithmic pressure to discuss the formation of delta-shocks for the Chaplygin gas equations. We solve the Riemann problem for the logarithmic perturbed equations by the analysis method in phase plane. The Riemann solutions with four different kinds of structures are obtained. We prove that as the logarithmic pressure vanishes, some Riemann solutions containing the two-shock tend to the delta-shock solution to the Chaplygin gas equations, the other Riemann solutions of the logarithmic perturbed equations tend to the two-contact discontinuity solution to the Chaplygin gas equations. Finally, we present some numerical results to exhibit the formation process of delta-shocks.