Formation of $\delta$-Shocks and Vacuum States in the Vanishing Pressure Limit of Solutions to the Euler Equations for Isentropic Fluids

Abstract

The phenomena of concentration and cavitation and the formation of δ-shocks and vacuum states in solutions to the Euler equations for isentropic fluids are identified and analyzed as the pressure vanishes. It is shown that, as the pressure vanishes, any two-shock Riemann solution to the Euler equations for isentropic fluids tends to a δ-shock solution to the Euler equations for pressureless fluids, and the intermediate density between the two shocks tends to a weighted δmeasure that forms the δ-shock. By contrast, any two-rarefaction-wave Riemann solution of the Euler equations for isentropic fluids is shown to tend to a two-contact-discontinuity solution to the Euler equations for pressureless fluids, whose intermediate state between the two contact discontinuities is a vacuum state, even when the initial data stays away from the vacuum. Some numerical results exhibiting the formation process of δ-shocks are also presented.

Linearization of the Riemann problem for a triangular system of conservation laws and delta shock wave formation process

“…to take ε neighborhood of the discontinuity line and to dispose characteristics in that neighborhood in a way that they do not intersect each other, and as ε → 0 all of them lump together into the discontinuity line. Of course, this will not be the standard characteristics for problem (1), (3). Nevertheless, along them approximate solution to our problem will remain constant.…”

“…This Riemann problem is intensively investigated in recent years [7,10,11,13,14,16,17,18,19,21,24,28,30,31,32,36,37,39] (the list is far from being complete). The reason for this lies in applicability of the system -it arises from (generalized) pressureless gas dynamics [3,27]. Another, purely mathematical reason, is the fact that under the following assumptions on f and g (see e.g.…”

Vanishing viscosity limit for Riemann solutions to zero-pressure gas dynamics with flux perturbation

“…As 1 = 0, system (1.6) is nothing but the Euler equations of isentropic gas dynamics with pressure perturbation. By using the vanishing pressure limit method, Chen and Liu [34] identified the stability of the delta shock wave of (1.1) under the pressure perturbation, which was equivalent to the formation of delta shock waves and vacuum states in solutions of system (1.6) as 2 → 0. Further, in [35] they also studied vanishing pressure limit of solutions to the nonisentropic fluids.…”

The Delta Standing Wave Solution for the Linear Scalar Conservation Law With Discontinuous Coefficients Using a Self-Similar Viscous Regularization

“…Special attentions were also paid in [12,16,17,24,31,33,35,37,38] to the formation of the delta shock waves in the Riemann solutions for some hyperbolic systems of conservation laws. There exist numerous excellent papers for the related equations and results about the measure-valued solutions such as the delta shock wave for hyperbolic systems of conservation laws, see [6,15,18,23,25,26] for instance.…”