On the delta shockwave interactions for the isentropic Chaplygin gas system consisting of three scalar equations

Abstract

The Riemann problem for the one-dimensional version of isentropic compressible Euler system for the Chaplygin gas consisting of three scalar equations is considered. It is shown that the Riemann solutions involve only two situations: the combination of three contact discontinuities or a delta shock wave. The generalized Rankine-Hugoniot conditions of delta shock wave are derived and the exact delta shock wave solution including the strength and propagation speed is obtained explicitly. The solutions to the perturbed Riemann problem are constructed globally when the initial data are taken to be the three piecewise constant initial data. The wave interaction problem is extensively investigated and some interesting phenomena are observed. It is shown that the limits of solutions to the perturbed Riemann problem converge to the corresponding ones to the Riemann problem when the perturbation parameter tends to zero. (Jie Xin) J 1 (ρ 3 , u 3 , v + ) J 7 0 J 9 0 J 10 0 J 8 0 J 6 0 J 8 0 J 5 0 J 7 0 J 2 0 J 3 0 J 4

Concentration and Cavitation in the Vanishing Pressure Limit of Solutions to a 3 × 3 Generalized Chaplygin Gas Equations

“…) the projection curves of the classical wave curves (3.7), (3.8), (3.13) and (3.15) in the (u, ρ)-phase plane, respectively. Then, by a similar analysis as in [24,25,28], we know that du dρ < 0 for…”

“…Noting that the state variables ρ and u are invariant and only the state variable v varies when across J, the slip line. Thus, motivated by [19,20,25], we are able to consider the elementary wave curves projected onto the upper-half (u, ρ)-phase plane. Here, the positions of ρ and u are exchanged in the phase plane just for convenience.…”

“…Motivated by [19,20,24,25,28], etc., the solution for (1.1) with delta distribution at the jump (i.e., the delta-shock solution) should be constructed, which is, under the Definition 2.1,…”

“…Nowadays, the delta shock wave has become a hot topic in the study of hyperbolic conservation laws. Besides, it is worth recalling that, although the delta shock wave is also obtained in solutions of (1.1) and (1.2) for CG (see [12,25]), the occurrence mechanism of which is essentially different from that of the PGD model (1.4). For the former, the system is strict hyperbolic and owns three kinds of degenerated characteristics.…”

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

“…That is to say, Riemann solutions of (1.2) and (1.3) are stable with respect to certain small perturbation of initial condition given by (1.4) where ε is regarded as the so-called perturbation parameter. It should be stressed that the initial condition taken in the form (1.4) has been intensively used to investigate the problem of wave interactions [32][33][34][35][36][37][38] for various hyperbolic systems of conservation laws.…”