  We shall deal with both the barotropic and the full compressible Euler system
in multiple space dimensions. Both systems are particular examples of
hyperbolic conservation laws. Whereas for scalar conservation laws there exists
a well-known complete well-posedness theory, and for one-dimensional systems
one also has achieved several results on existence and uniqueness, in the case
of multi-dimensional systems there are even negative results regarding
uniqueness: With the so-called convex integration method it is possible to show
that there exist initial data for which the compressible Euler equations in
multiple space dimensions admit infinitely many solutions. The convex
integration technique was originally developed in the context of differential
inclusions and has later been applied in groundbreaking papers by De Lellis and
Sz\'ekelyhidi to the incompressible Euler equations which led to infinitely
many solutions. In the literature this result has been refined in order to
obtain solutions for the compressible Euler system as well. The common feature
of all of these non-uniqueness results for compressible Euler is an ansatz
which reduces the compressible Euler equations to some kind of "incompressible
system" for which a slight modification of the incompressible theory can be
applied. In this work we present a first result of a direct application of
convex integration to the barotropic compressible Euler equations. With the
help of this result we will show existence of initial data for which there are
infinitely many solutions both for the barotropic and full Euler system.
