The Vanishing Pressure Limit of Riemann Solutions to the Non-Isentropic Euler Equations for Generalized Chaplygin Gas

Qixia Ding,Lihui Guo

doi:10.1155/2019/5253717

Abstract

We analyze the appearance of delta shock wave and vacuum state in the vanishing pressure limit of Riemann solutions to the non-isentropic generalized Chaplygin gas equations. As the pressure vanishes, the Riemann solution including two shock waves and possible one contact discontinuity converges to a delta shock wave solution. Both the densityρand the internal energyHsimultaneously present a Dirac delta singularity. And the Riemann solution involving two rarefaction waves and possible one contact discontinuity converges to a solution involving vacuum state of the transport equations.

Highlights

The compressible Euler equations of non-isentropic fluids in one dimension are given by ρt +x = 0,t + x = (1)
We analyze the appearance of delta shock wave and vacuum state in the vanishing pressure limit of Riemann solutions to the non-isentropic generalized Chaplygin gas equations
In one-dimensional case, we need to prove the following conclusions: (1) For the case of u+ < u−, Riemann solution of system (1)–(2) tends to a δ-shock wave solution when ε reduces to a certain value ε2, and the δ-shock wave tends to a δ-shock solution of zero-pressure flow (4) as ε 󳨀→ 0

Summary

Introduction

We analyze the appearance of delta shock wave and vacuum state in the vanishing pressure limit of Riemann solutions to the non-isentropic generalized Chaplygin gas equations. In 2013, Wang [5] studied the generalized Chaplygin equations with constant initial data and obtained the global Riemann solution involving nonclassical wave (delta shock wave). He obtained the Riemann solutions involving vacuum state, delta shock wave, and contact discontinuities.

Results

Conclusion