On action rate admissibility criteria

TL;DR

This work tackles nonuniqueness in weak solutions for the isentropic Euler equations by introducing a local-in-time least action criterion $LAAP_0$, which uses the rate of change of action to select among competing solutions. Building on convex integration and fan-subsolution constructions, the authors show that for the 2D Riemann problem there exist parameter regimes where the classical $2$-shock is strictly $LAAP_0$-admissible and thus uniquely preferred over both convex integration wild solutions and recently constructed hybrids. The results hold rigorously for the case $p(\rho)=\rho^2$ (i.e., $\gamma=2$) and, supported by numerical evidence, extend to $\gamma\in[1,3]$, providing a robust, local criterion that complements entropy-based selections and excludes globally nonlocal pathologies. This establishes a practical selection principle rooted in the least action that preserves locality in time while excluding problematic global constructions.

Abstract

We formulate new admissibility criteria for initial value problems motivated by the least action principle. These are applied to a two-dimensional Riemann initial value problem for the isentropic compressible Euler fluid flow. It is shown that the criterion prefers the 2-shock solution to solutions obtained by convex integration by Chiodaroli and Kreml or to the hybrid solutions recently constructed by Markfelder and Pellhammer.

Figures (1)

• Figure 1: Domains of integration

Theorems & Definitions (13)

• Definition 2.1: LAAP
• Definition 2.2
• Remark 2.1
• Definition 2.3
• Remark 2.2
• Proposition 2.1
• Definition 3.1: Fan sub-solution
• Lemma 3.1
• Remark 3.1: Remarks on Lemma \ref{['corru']}
• Theorem 5.1