A General Convex Integration Scheme for the Isentropic Compressible Euler Equations

TL;DR

This paper introduces a convex integration method for the isentropic Euler equations that constructs weak solutions from subsolutions, extending previous incompressible schemes and simplifying the perturbation process.

Contribution

It develops a general convex integration framework for the isentropic Euler equations, allowing construction of weak solutions by perturbing only momenta, not densities.

Findings

Established a convex integration scheme for isentropic Euler equations.

Characterized the $ ext{Lambda}$-convex hull of the constitutive set.

Facilitated the construction of weak solutions from subsolutions.

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.