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

Simon Markfelder, Valentin Pellhammer

TL;DR

This work addresses the selection problem for multi-dimensional barotropic Euler solutions by testing the recently proposed least action admissibility principle. Using convex integration and a careful gluing construction, the authors build a family of admissible weak solutions to a 2D Riemann problem that includes a central oscillatory region and a downward energy jump, allowing them to compare the action of a canonical 1-D solution with a nontrivial convex integration solution. They explicitly compute action densities and show that, for suitable Riemann data and time horizon $T$, the convex integration solution has strictly smaller action than the 1-D solution, thereby violating the least action principle in this setting. This counterexample demonstrates that the least action admissibility principle cannot reliably select the physically relevant solution for the Euler equations and raises questions about the role of such variational criteria in turbulent multi-dimensional flows.

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.

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

TL;DR

This work addresses the selection problem for multi-dimensional barotropic Euler solutions by testing the recently proposed least action admissibility principle. Using convex integration and a careful gluing construction, the authors build a family of admissible weak solutions to a 2D Riemann problem that includes a central oscillatory region and a downward energy jump, allowing them to compare the action of a canonical 1-D solution with a nontrivial convex integration solution. They explicitly compute action densities and show that, for suitable Riemann data and time horizon , the convex integration solution has strictly smaller action than the 1-D solution, thereby violating the least action principle in this setting. This counterexample demonstrates that the least action admissibility principle cannot reliably select the physically relevant solution for the Euler equations and raises questions about the role of such variational criteria in turbulent multi-dimensional flows.

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.

Paper Structure

This paper contains 7 sections, 5 theorems, 43 equations, 2 figures.

Table of Contents

  1. Introduction and main result
  2. Shape of the solutions constructed in this paper
  3. Proof of Theorem \ref{['thm:mainTheorem']}
  4. Special Riemann data and the corresponding 1-D solution
  5. Construction of another solution
  6. Comparison of action
  7. Concluding remarks

Key Result

Theorem 1.4

For each $T>0$, there existIn fact, such Riemann data $(\varrho_\pm,\mathbf{u}_\pm)$ are independent of $T>0$. Riemann initial data $(\varrho_\pm,\mathbf{u}_\pm)\in\mathbb{R}^+\times\mathbb{R}^2$ such that the 1-D solution to eq:euler, eq:RiemannData does not fulfill the least action admissibility p

Figures (2)

  • Figure 1: The solution $(\varrho_{{\rm ex}},\mathbf{u}_{{\rm ex}})$ (black), which is constructed in Section \ref{['subsec:constructionCounterEx']}, and the 1-D solution $(\varrho_{1d},\mathbf{u}_{1d})$ (red) in the $y-t$ plane for $T_0=\tfrac{1}{2}$.
  • Figure 2: The functions $A[\varrho,\mathbf{u}]$ and $\widetilde{\mathcal{A}}[\varrho,\mathbf{u}]$ for the solutions $(\varrho_{1d},\mathbf{u}_{1d})$ (red) and $(\varrho_{{\rm ex}},\mathbf{u}_{{\rm ex}})$ (black), respectively, where $T=1$.

Theorems & Definitions (14)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 4 more