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<γ⩽1+2n. This result can be regarded as a compressible counterpart of the one obtained by Székelyhidi–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.