The Aw–Rascle traffic model with Chaplygin pressure

In this paper, we introduce the Chaplygin pressure function to the Aw–Rascle (AR) traffic model. The solutions to the Riemann problem of the AR model with Chaplygin pressure are obtained constructively. In particular, the existence and uniqueness of delta shock solutions are established under the generalized Rankine–Hugoniot relation and entropy conditions. The delta shock may be useful for the description of the serious traffic jam. More importantly, it is proved that the limits of the Riemann solutions of the AR traffic model with Chaplygin pressure are exactly those of the pressureless gas dynamics model as the traffic pressure vanishes. Furthermore, we study the initial–boundary value problem of the AR model with Chaplygin pressure. A boundary entropy condition is derived, thanks to the results of Dubois and Le Floch [F. Dubois, P. Le Floch, Boundary conditions of nonlinear hyperbolic systems of conservation laws, Journal of Differential Equations 71 (1988) 93–122]. The application of our results in real traffic problem is also given.