Concentration in vanishing adiabatic exponent limit of solutions to the Aw-Rascle traffic model

In this paper, we study the phenomenon of concentration and the formation of delta shock wave in vanishing adiabatic exponent limit of Riemann solutions to the Aw-Rascle traffic model. It is proved that as the adiabatic exponent vanishes, the limit of solutions tends to a special delta-shock rather than the classical one to the zero pressure gas dynamics. In order to further study this problem, we consider a perturbed Aw-Rascle model and proceed to investigate the limits of solutions. We rigorously proved that, as the adiabatic exponent tends to one, any Riemann solution containing two shock waves tends to a delta-shock to the zero pressure gas dynamics in the distribution sense. Moreover, some representative numerical simulations are exhibited to confirm the theoretical analysis.

Automated summary

This paper studies the concentration phenomenon and the formation of delta shock wave in the vanishing adiabatic exponent limit of Riemann solutions to the Aw-Rascle traffic model. It is proven that as the adiabatic exponent tends to zero, the limit of solutions tends to a special delta-shock rather than the classical one in zero pressure gas dynamics. By considering a perturbed Aw-Rascle model, it is rigorously proved that as the adiabatic exponent tends to one, any Riemann solution containing two shock waves tends to a delta-shock to the zero pressure gas dynamics. Representative numerical simulations are also presented to confirm the theoretical analysis.