Delta Shocks and Vacuum States in the Vanishing Pressure Limit of Riemann Solutions to the Isentropic Euler System for Chaplygin Gas

In this paper, we study the formation of delta shock waves and vacuum states of Riemann problem for isentropic Euler system with Chaplygin pressure by vanishing pressure limit method under the self-similar coordinate. The state equation is p = -epsilon(2)/rho. There is no vacuum state initially. We proved that as epsilon -> 0, the self-similar solution can be divided into three cases which may contains a contact discontinuity, a vacuum state or a delta shock wave. Moreover, we proved that for Chaplygin gas, there exists a critical value epsilon(1) > 0 depending on the initial data, such that for any epsilon > epsilon(1), there is no delta shock wave in the solution. For epsilon is an element of (0, epsilon(1)] Ell and for suitable initial data, the solution contains a delta shock wave which can be expressed explicitly. As epsilon -> 0, the sequence of delta shock wave solutions converge to the delta shock wave solution of the transport equations with the same initial data.