Limit of Riemann solutions to the nonsymmetric system of Keyfitz-Kranzer type

The limit of Riemann solutions to the nonsymmetric system of Keyfitz-Kranzer type with a scaled pressure is considered for both polytropic gas and generalized Chaplygin gas. In the former case, the delta shock wave can be obtained as the limit of shock wave and contact discontinuity when u - > u + and the parameter ϵ tends to zero. The point is, the delta shock wave is not the one of transport equations, which is obviously different from cases of some other systems such as Euler equations or relativistic Euler equations. For the generalized Chaplygin gas, unlike the polytropic or isothermal gas, there exists a certain critical value ϵ 2 depending only on the Riemann initial data, such that when ϵ drops to ϵ 2, the delta shock wave appears as u - > u +, which is actually a delta solution of the same system in one critical case. Then as ϵ becomes smaller and goes to zero at last, the delta shock wave solution is the exact one of transport equations. Furthermore, the vacuum states and contact discontinuities can be obtained as the limit of Riemann solutions when u - < u + and u - = u +, respectively.