The Least Action Admissibility Principle

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.

16 pages, ancillary Maple code verifies proof of Theorem 6, accepted for publication in Archive for Rational Mechanics and Analysis

References in corpus (6)

• On admissibility criteria for weak solutions of the Euler equations
• Global ill-posedness of the isentropic system of gas dynamics
• On the energy dissipation rate of solutions to the compressible isentropic Euler system
• Maximal dissipation and well-posedness for the compressible Euler system
• On a degenerate elliptic problem arising in the least action principle for Rayleigh-Taylor subsolutions
• Non-Uniqueness in Plane Fluid Flows

Cited by in corpus (2)

• Maximal turbulence as a selection criterion for measure-valued solutions
• The Rayleigh-Taylor instability with local energy dissipation