Global ill-posedness for a dense set of initial data to the Isentropic system of gas dynamics

Abstract

In dimension $n=2$ and $3$, we show that for any initial datum belonging to a dense subset of the energy space, there exist infinitely many global-in-time admissible weak solutions to the isentropic Euler system whenever $1<\gamma \le 1+\frac{2}{n}$. This result can be regarded as a compressible counterpart of the one obtained by Szekelyhidi--Wiedemann (ARMA, 2012) for incompressible flows. Similarly to the incompressible result, the admissibility condition is defined in its integral form. Our result is based on a generalization of a key step of the convex integration procedure. This generalization allows, even in the compressible case, to convex integrate any smooth positive Reynolds stress. A large family of subsolutions can then be considered. These subsolutions can be generated, for instance, via regularization of any weak inviscid limit of an associated compressible Navier--Stokes system with degenerate viscosities.

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References in corpus (9)

• On admissibility criteria for weak solutions of the Euler equations
• Existence of Global Weak Solutions for 3D Degenerate Compressible Navier-Stokes Equations
• Young Measures Generated by Ideal Incompressible Fluid Flows
• On the weak solutions to the equations of a compressible heat conducting gas
• Maximal dissipation and well-posedness for the compressible Euler system
• h-Principles for the incompressible Euler equations
• Non-uniqueness of admissible weak solutions to the Riemann problem for the isentropic Euler equations
• Formation of shocks for 2D isentropic compressible Euler
• Non-uniqueness of admissible weak solution to the Riemann problem for the full Euler system in 2D

Cited by in corpus (2)

• On Non-uniqueness of continuous entropy solutions to the isentropic compressible Euler equations
• On bounded two-dimensional globally dissipative Euler flows