The local least action criterion fails as a selection criterion for weak solutions of the compressible Euler equations

Abstract

It is now well known that the classical notion of admissible weak solutions (also known as weak entropy solutions) does not restore uniqueness for the multi-dimensional compressible Euler equations. Indeed, convex integration has shown that admissible weak solutions are in general highly non-unique. This has motivated additional selection criteria intended to rule out the counterintuitive solutions generated by convex integration. In this paper we prove that the local least action criterion introduced by H.~Gimperlein, M.~Grinfeld, R.~J.~Knops and M.~Slemrod does not serve as a proper selection criterion, since it fails to select the solution that is intuitively expected to be physically relevant.