Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw–Rascle model

A traffic flow model describing the formation and dynamics of traffic jams was introduced by Berthelin et al., which consists of a constrained pressureless gas dynamics system and can be derived from the Aw–Rascle model under the constraint condition ρ⩽ρ⁎ by letting the traffic pressure vanish. In this paper, we give up this constraint condition and consider the following form{ρt+(ρu)x=0,(ρu+εp(ρ))t+(ρu2+εup(ρ))x=0, in which p(ρ)=ργ with γ>1. The formal limit of the above system is the pressureless gas dynamics system in which the density develops delta-measure concentration in the Riemann solution. However, the propagation speed and the strength of the delta shock wave in the limit situation are different from the classical results of the pressureless gas dynamics system with the same Riemann initial data. In order to solve it, the perturbed Aw–Rascle model is proposed as{ρt+(ρu)x=0,(ρu+εγp(ρ))t+(ρu2+εup(ρ))x=0, whose behavior is different from that of the Aw–Rascle model. It is proved that the limits of the Riemann solutions of the perturbed Aw–Rascle model are exactly those of the pressureless gas dynamics model.