Non-uniqueness of delta shocks and contact discontinuities in the multi-dimensional model of Chaplygin gas

TL;DR

This paper demonstrates the non-uniqueness of solutions, including delta shocks and contact discontinuities, for the multi-dimensional Chaplygin gas model, challenging classical solution selection criteria.

Contribution

It proves the existence of infinitely many admissible weak solutions for the 2D Euler equations with Chaplygin gas, even for initial data with classical solutions.

Findings

Existence of infinitely many solutions for certain initial data.

Non-uniqueness persists even with classical contact discontinuities.

Maximal dissipation principle does not select the classical solution.

Abstract

We study the Riemann problem for the isentropic compressible Euler equations in two space dimensions with the pressure law describing the Chaplygin gas. It is well known that there are Riemann initial data for which the 1D Riemann problem does not have a classical $BV$ solution, instead a $\delta$-shock appears, which can be viewed as a generalized measure-valued solution with a concentration measure in the density component. We prove that in the case of two space dimensions there exists infinitely many bounded admissible weak solutions starting from the same initial data. Moreover, we show the same property also for a subset of initial data for which the classical 1D Riemann solution consists of two contact discontinuities. As a consequence of the latter result we observe that any criterion based on the principle of maximal dissipation of energy will not pick the classical 1D solution as the physical one.