On the $Λ$-Convex Hull for Convex Integration Applied to the Isentropic Compressible Euler System

Simon Markfelder

Published 2020-01-13Version 1

We consider the isentropic compressible Euler equations in multiple space dimensions. In the past it has been shown via convex integration that this system allows for infinitely many solutions. However all non-uniqueness results available in the literature were achieved by reducing the equations to some kind of incompressible system and applying convex integration to the latter. This ansatz seems to be quite restrictive concerning the solutions that are obtained and also concerning the initial data for which the method works. A direct application of convex integration to the compressible Euler equations could overcome these restrictions. This paper can be viewed as the first step towards such a direct application as we present a new setup for convex integration and compute the corresponding $\Lambda$-convex hull.