Failure of the least action admissibility principle in the context of the compressible Euler equations

TL;DR

This paper demonstrates that the least action admissibility principle fails to select physically relevant solutions for the compressible Euler equations, challenging its suitability as a solution criterion.

Contribution

The paper critically evaluates the least action admissibility principle, showing it excludes physically relevant solutions and questioning its validity for the Euler equations.

Findings

Least action principle rules out physically relevant solutions

Convex integration reveals non-uniqueness of weak solutions

The principle may need reconsideration or rejection

Abstract

Finding a proper solution concept for the multi-dimensional barotropic compressible Euler equations and related systems is still an unsolved problem. As revealed by convex integration, the classical notion of an admissible weak solutions (also known as weak entropy solutions) does not lead to uniqueness and allows for solutions which do not seem to be physical. For this reason, people have studied additional criteria in view of their ability to rule out the counterintuitive solutions generated by convex integration. Recently, in [H.~Gimperlein, M.~Grinfeld, R.~J.~Knops and M.~Slemrod: The least action admissibility principle, arXiv: 2409.07191 (2024)] it was suggested that the least action admissibility principle serves as the desired selection criterion. In this paper, however, we show that the least action admissibility principle rules out the solution which is intuitively the physically relevant one. Consequently, one either has to reconsider one's intuition, or the least action admissibility principle must be discarded.