The Perturbed Riemann Problem for a Macroscopic Production Model with Chaplygin Gas

Abstract

The exact Riemann solutions for a macroscopic production model under the equation of state given by the Chaplygin gas are solved explicitly for all possible Riemann initial data. It is discovered interestingly that a composite hyperbolic wave is involved in Riemann solution under some specially designated initial conditions, which is made up of a rarefaction wave and a delta contact discontinuity attached on the wave front of the rarefaction wave. Furthermore, the constructions of global solutions to the perturbed Riemann problem for this system are also displayed in completely explicit forms when the initial data are taken to be three piecewise constant states under some suitable restrictive conditions by using the method of characteristics. During the process of constructing global solutions, the interactions of elementary waves are studied in detail. Moreover, it is proved rigorously that Riemann solutions are stable with respect to the specific small perturbations of Riemann initial data.

The adiabatic exponent limits of Riemann solutions for the extended macroscopic production model

“…Furthermore, some new exact solutions for (1.1) in the case of p(ρ) = ρ have also been obtained by Sil and Raja Sekhar [37,38] with the help of nonclassical symmetry analysis as well as interactions of nonlinear waves have been widely investigated by Minhajul and Raja Sekhar [21]. Very recently, exact Riemann solutions have been obtained and wave interactions have been dealt with carefully in [43] for (1.1) by taking the pressure-density relation p(ρ) = − 1 ρ for Chaplygin gas. It should be addressed especially that the asymptotic limits ε → 0 of Riemann solutions for the model (1.1) with the perturbed anticipated factor p(ρ) = ερ have been considered by Zhang and Sun [47].…”