Some New Nonlinear Hyperbolic Waves in Macroscopic Production Model

In this paper, the Riemann solutions and their asymptotic limits for the macroscopic production model with modified Chaplygin gas are investigated. Due to this model belonging to the Temple class and the curve of this rarefaction wave lying between the one of shock and uncertain waves in the phase plane, the Riemann solutions we constructed differ from those of previous studies. More specifically, a special structure $R\bar{J}+J$ is found in the construction of the Riemann solution. $R\bar{J}$ is formed by a rarefaction wave $R$ and a discontinuity $\bar{J}$ attached to the wavefront of this rarefaction wave. It should be stressed that the rarefaction wave and the shock wave share the same wave velocity on this discontinuity $\bar{J}$. In the asymptotic limits of the Riemann solutions, not all solutions converge to those of the conservation law equations when the perturbation parameters tend to 0 with the same initial data. What is interesting is that the vacuum and delta shock wave solutions are obtained by the small Riemann solution $S+J$, which consists of a shock wave $S$ followed by a contact discontinuity $J$, where the small Riemann solution means that the solution whose structure remains unchanged for the solution of the Riemann problem of the macroscopic production model with modified Chaplygin gas under the small enough perturbation parameters.