The Vanishing Pressure Limits of Riemann Solutions to the Chaplygin Euler Equations

TL;DR

This paper investigates the behavior of Riemann solutions to Chaplygin Euler equations as pressure approaches zero, revealing convergence to delta shocks, contact discontinuities, and vacuum states, with detailed analysis of shock and wave dynamics.

Contribution

It provides a detailed analysis of the vanishing pressure limits of Riemann solutions for Chaplygin Euler equations, highlighting differences from polytropic or isothermal gases.

Findings

Shock wave solutions converge to delta shocks with detailed strength and speed analysis.

Rarefaction wave solutions tend to contact discontinuities and vacuum states.

Distinct behaviors from classical gas models are identified in the zero-pressure limit.

Abstract

The Riemann solutions to Chaplygin Euler equations with a scaled pressure are considered. When the pressure vanishes, there are three cases. The Riemann solution containing two shock waves converges to the delta shock wave solution of the transport equations. During this process, both the strength and propagation speed of the delta shock are investigated in detail. We find that there is something different from that for polytropic or isothermal gas. The Riemann solution containing two rarefaction waves tends to the two contact discontinuity solution to the transport equations as the pressure goes to zero. The intermediate state between the two contact discontinuities is a vacuum state. The Riemann solution containing one rarefaction wave and one shock wave tends to the contact discontinuity solution to transport equations as the pressure vanishes.