The Least Action Admissibility Principle

TL;DR

This paper introduces a new admissibility criterion based on the least action principle for selecting physically relevant weak solutions in continuum mechanics, especially in cases of non-uniqueness, and compares it with entropy criteria.

Contribution

It proposes the least action admissibility principle as a novel criterion for weak solutions and demonstrates its effectiveness in selecting classical solutions over convex integration solutions.

Findings

The least action principle favors classical shock solutions over convex integration solutions.

Dafermos's entropy criterion prefers convex integration solutions.

For pressure p(ρ)=ρ^2, the two shock solution is always preferred when convex solutions exist.

Abstract

This paper provides a new admissibility criterion for choosing physically relevant weak solutions of the equations of Lagrangian and continuum mechanics when non-uniqueness of solutions to the initial value problem occurs. The criterion is motivated by the classical least action principle but is now applied to initial value problems which exhibit non-unique solutions. Examples are provided to Lagrangian mechanics and the Euler equations of barotropic fluid mechanics. In particular, we show the least action admissibility principle prefers the classical two shock solution to the Riemann initial value problem to certain solutions generated by convex integration. On the other hand, Dafermos's entropy criterion prefers convex integration solutions to the two shock solutions. Furthermore, when the pressure is given by $p(\rho)=\rho^2$, we show that the two shock solution is always preferred whenever the convex integration solutions are defined for the same initial data.