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In this paper we establish a new convex integration approach for thebarotropic compressible Euler equations in two space dimensions. In contrast toexisting literature, our new method generates not only the momentum for givendensity, but also the energy and the energy flux. This allows for a simple wayto construct admissible solutions, i.e. solutions which satisfy the energyinequality.  Moreover using the convex integration method developed in this paper, we showthat the local maximal dissipation criterion fails in the following sense:There exist wild solutions which beat the self-similar solution of theone-dimensional Riemann problem extended to two dimensions. Hence the localmaximal dissipation criterion rules out the self-similar solution.  The convex integration machinery itself is carried out in a very general way.Hence this paper provides a general framework for convex integration which noteven specifies the form of the partial differential equations underconsideration. Therefore this general framework is applicable in many differentsituations.

A New Convex Integration Approach for the Compressible Euler Equations and Failure of the Local Maximal Dissipation Criterion