Theoretical analysis reveals that the local least action criterion fails to select physically relevant solutions in compressible Euler equations, indicating its inadequacy as a selection rule.
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.
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Markfelder et al. (2026) studied this question.
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