The adiabatic exponential limits of Riemann solutions in the isentropic three-component model

Abstract

The explicit construction of Riemann solutions is achieved for an ideally isentropic three-component model owning a unitary velocity and a collective pressure in one space dimension under the hypotheses without mass and heat transfer and also without viscosity. In addition, the asymptotic results of Riemann solutions are explored at length by sending the adiabatic exponent drop to one. On the one side, it reveals the concentration phenomenon, where the Riemann solution with a 1-shock, 2,3-contact and 4-shock waves converges to a delta shock solution. On the other side, it also exhibits the cavitation phenomenon, where all internal states in the 1-rarefaction and 4-rarefaction waves become vacuum ones by sending this limit. Finally, some representative numerical simulations are offered to observe the formation of delta shock wave and vacuum state in a more intuitive way as the adiabatic exponent tends to one, which is consistent with the theoretical analysis.