The Riemann problem for the multidimensional isentropic system of gas dynamics is ill-posed if it contains a shock

Abstract

In this paper we consider the isentropic compressible Euler equations in two space dimensions together with particular initial data. The latter consists only of two constant states, where one state lies on the lower and the other state on the upper half plane. The aim is to investigate if there exists a unique entropy solution or if the convex integration method produces infinitely many entropy solutions. For some initial states this question has been answered by E. Feireisl and O. Kreml, and also G.-Q. Chen and J. Chen, where there exists a unique entropy solution. For other initial states E. Chiodaroli, O. Kreml and C. De Lellis showed that there are infinitely many entropy solutions. For still other initial states the question on uniqueness remained open and this will be the content of this paper. This paper can be seen as a completion of the aforementioned papers by showing that the solution is non-unique in all cases (except if the solution is smooth).

References in corpus (4)

• On admissibility criteria for weak solutions of the Euler equations
• Global ill-posedness of the isentropic system of gas dynamics
• On the energy dissipation rate of solutions to the compressible isentropic Euler system
• Non-uniqueness of admissible weak solutions to the Riemann problem for the isentropic Euler equations

Cited by in corpus (14)

• Non-uniqueness of admissible weak solutions to the Riemann problem for the isentropic Euler equations
• Solution semiflow to the isentropic Euler system
• Vanishing viscosity limit to the planar rarefaction wave for the two-dimensional compressible Navier-Stokes equations
• Kolmogorov-Type Theory of Compressible Turbulence and Inviscid Limit of the Navier-Stokes Equations in ${\mathbb{R}}^3$
• Non-uniqueness of admissible weak solution to the Riemann problem for the full Euler system in 2D
• Shocks Make the Riemann Problem for the Full Euler System in Multiple Space Dimensions Ill-posed
• Uniqueness of a planar contact discontinuity for 3D compressible Euler system in a class of zero dissipation limits from Navier-Stokes-Fourier system
• The 2-d isentropic compressible Euler equations may have infinitely many solutions which conserve energy
• On Non-uniqueness of continuous entropy solutions to the isentropic compressible Euler equations
• A New Convex Integration Approach for the Compressible Euler Equations and Failure of the Local Maximal Dissipation Criterion
• Finite time stability for the Riemann problem with extremal shocks for a large class of hyperbolic systems
• On the density of wild initial data for the compressible Euler system
• The Rayleigh-Taylor instability with local energy dissipation
• Contact discontinuities for 2-D isentropic Euler are unique in 1-D but wildly non-unique otherwise