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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.

In this paper, we study the limiting behavior of Riemann solutions to the Euler equations of one-dimensional compressible fluid flow as $\gamma$ tends to one. We show that the limit solution forms the delta wave to the pressureless Euler system of one-dimensional compressible fluid flow in the distribution sense. Some numerical results exhibiting the phenomena of concentration are also presented., Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:1904.03462

In this paper, we investigate the vanishing adiabatic exponent limits of the Riemann solutions to the isentropic Euler equations for power law with a Coulomb-like friction term. The formation of delta shock waves and vacuum states is identified and analyzed as the adiabatic exponent vanishes. Unlike the homogeneous case, the Riemann solutions are no longer self-similar. We rigorously justify that, as the adiabatic exponent vanishes, the Riemann solutions to the isentropic Euler equations for power law with a Coulomb-like friction term converge to the Riemann solutions to the zero pressure gas dynamics model with a body force. Moreover, we also give some numerical results to confirm the theoretical analysis.