Abstract
We construct the solutions to the Riemann problem for the isentropic extended Chaplygin gas dynamic system for all kinds of situations by the phase plane analysis. We investigate the asymptotic limits of solutions to this problem in detail when the pressure given by the state equation of the system becomes the one of pressureless gas. During the process of vanishing pressure, the two-shock Riemann solution tends to a delta shock solution, whereas the two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution with a vacuum state.
Highlights
The Chaplygin gas model was first proposed by Chaplygin [1] as a model in aerodynamics with the equation of state in the form p = –B/ρ with constant B > 0
The formation process of δ-shock wave and vacuum state can be observed when we consider the limit of solutions to the Riemann problem for the isentropic extended Chaplygin gas dynamic system (1.1) as the vanishing pressure Ak (k = 1, . . . , n), B → 0
[36], we have considered the limiting relations of Riemann solutions from the isentropic extended Chaplygin gas dynamic system to the isentropic Chaplygin gas dynamic system when only the limit Ak (k = 1, . . . , n) → 0 is taken in system (1.1)
Summary
The formation process of δ-shock wave and vacuum state can be observed when we consider the limit of solutions to the Riemann problem for the isentropic extended Chaplygin gas dynamic system (1.1) as the vanishing pressure Ak The formation of δ-shock wave and vacuum state to the Riemann problem (1.2) and (1.3) was considered initially for the isothermal [21] and isentropic [10] situations by the vanishing pressure limit approach. 3, we investigate in detail the basic properties of the isentropic extended Chaplygin gas dynamic system (1.1) and construct the solutions to the Riemann problem (1.1) and (1.3) for all kinds of Riemann initial data. 4, we make a step further by investigating the limit relations of Riemann solutions from the isentropic extended Chaplygin gas dynamic system (1.1) to the pressureless gas dynamic system (1.2). The formation of δ-shock wave and vacuum state can be observed during the process of vanishing pressure limit
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