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The Least Action Admissibility Principle

Heiko Gimperlein, Michael Grinfeld, Robin J. Knops, Marshall Slemrod

TL;DR

This work introduces the Least Action Admissibility Principle (LAAP) as a selection mechanism for non-unique weak solutions of the Euler equations in barotropic fluids, using the action derived from a Lagrangian formulation to compare candidate motions. It systematically contrasts LAAP with Dafermos’ entropy-rate criterion, showing that LAAP can favor classical shock solutions over convex-integration constructions in several isentropic-Riemann settings, and demonstrates consistency with known non-uniqueness resolution phenomena such as vanishing viscosity limits. Through detailed analysis of the Dafermos oscillator, the barotropic Euler system, and the two-dimensional Riemann problem, the authors establish conditions under which LAAP prefers the two-shock solution and, in the special case $p(\rho)=\rho^2$, obtains a global result favoring the two-shock solution over convex integration. The results illuminate how action-based admissibility interacts with energy-entropy criteria, offering a complementary tool for selecting physically relevant weak solutions in continuum mechanics.

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 $p(ρ)=ρ^2$, we show that the two shock solution is always preferred whenever the convex integration solutions are defined for the same initial data.

The Least Action Admissibility Principle

TL;DR

This work introduces the Least Action Admissibility Principle (LAAP) as a selection mechanism for non-unique weak solutions of the Euler equations in barotropic fluids, using the action derived from a Lagrangian formulation to compare candidate motions. It systematically contrasts LAAP with Dafermos’ entropy-rate criterion, showing that LAAP can favor classical shock solutions over convex-integration constructions in several isentropic-Riemann settings, and demonstrates consistency with known non-uniqueness resolution phenomena such as vanishing viscosity limits. Through detailed analysis of the Dafermos oscillator, the barotropic Euler system, and the two-dimensional Riemann problem, the authors establish conditions under which LAAP prefers the two-shock solution and, in the special case , obtains a global result favoring the two-shock solution over convex integration. The results illuminate how action-based admissibility interacts with energy-entropy criteria, offering a complementary tool for selecting physically relevant weak solutions in continuum mechanics.

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 , we show that the two shock solution is always preferred whenever the convex integration solutions are defined for the same initial data.
Paper Structure (5 sections, 9 theorems, 56 equations, 2 figures)

This paper contains 5 sections, 9 theorems, 56 equations, 2 figures.

Table of Contents

  1. The Least Action Principle
  2. The Dafermos oscillator
  3. The barotropic compressible Euler system
  4. Convex integration and the Riemann problem
  5. Global result for $p(\rho) = \rho^2$

Key Result

Theorem 1

Let $n \geq 2$, $\Omega \subset {\mathbb R}^n$ a bounded open set, $T>0$ and $\Omega' \supset \supset \Omega$ locally Lipschitz. For a positive constant $\overline{\rho}$, assume that $\rho_0 \in C^1({\mathbb R}^n)$ is a positive function satisfying $\rho_0 ({\boldsymbol x}) = \overline{\rho} > 0$ f Then there exists a bounded initial momentum ${\boldsymbol m}_0$ with $\hbox{\em supp}\, ({\boldsym

Figures (2)

  • Figure 1: A quadratic potential $U$.
  • Figure 2: Behavior of $\chi (t)$.

Theorems & Definitions (13)

  • Theorem 1: AkramovWiedemann
  • Theorem 2: AkramovWiedemann
  • Corollary 3: AkramovWiedemann
  • Lemma 4: Lemma 3.2 of ChioDeLKr
  • Theorem 5
  • Theorem 6
  • Proposition 7
  • proof
  • Proposition 8
  • proof
  • ...and 3 more