A General Convex Integration Scheme for the Isentropic Compressible Euler Equations

Abstract

We prove via convex integration a result that allows to pass from a so-called subsolution of the isentropic Euler equations (in space dimension at least $2$) to exact weak solutions. The method is closely related to the incompressible scheme established by De Lellis--Sz\'ekelyhidi, in particular we only perturb momenta and not densities. Surprisingly, though, this turns out not to be a restriction, as can be seen from our simple characterization of the $\Lambda$-convex hull of the constitutive set. An important application of our scheme will be exhibited in forthcoming work by Gallenm\"uller--Wiedemann.