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Oscillatory approximations and maximum entropy principle for the Euler system of gas dynamics

Eduard Feireisl, Mária Lukáčová-Medvid'ová, Changsheng Yu

TL;DR

The paper addresses the nonuniqueness of the Euler system by examining measure-valued (DMV) solutions generated by oscillatory sequences of consistent approximations. It develops a rigorous DMV framework, proves that maximal DMV solutions concentrate as Dirac masses (hence are weak solutions), and shows that limits of oscillatory consistent approximations cannot be maximal computable DMV solutions, aligning with Dafermos' criterion only in non-oscillatory cases. Through a blend of analytic results (concatenation, energy/entropy defect analysis) and numerical experiments, the authors demonstrate that standard schemes yield DMV limits with nonzero energy/entropy defects, challenging the universality of entropy-maximizing admissibility. The work introduces energy-preserving DMV formulations, extends maximality results to these settings, and suggests a conjecture linking strong convergence to maximal computable DMV solutions and oscillatory behavior to nonexistence of a maximal computable DMV limit. Altogether, the results illuminate when entropy-based selection principles are consistent with measure-valued limits and have implications for numerical design and interpretation of Euler-system approximations.

Abstract

We show that the measure-valued solutions of the Euler system of gas dynamics generated by oscillatory sequences of consistent approximations violate the principle of maximal entropy production formulated by Dafermos. Numerical results illustrate that solutions obtained by standard numerical methods may be oscillatory and thus do not comply with the Dafermos criterion.

Oscillatory approximations and maximum entropy principle for the Euler system of gas dynamics

TL;DR

The paper addresses the nonuniqueness of the Euler system by examining measure-valued (DMV) solutions generated by oscillatory sequences of consistent approximations. It develops a rigorous DMV framework, proves that maximal DMV solutions concentrate as Dirac masses (hence are weak solutions), and shows that limits of oscillatory consistent approximations cannot be maximal computable DMV solutions, aligning with Dafermos' criterion only in non-oscillatory cases. Through a blend of analytic results (concatenation, energy/entropy defect analysis) and numerical experiments, the authors demonstrate that standard schemes yield DMV limits with nonzero energy/entropy defects, challenging the universality of entropy-maximizing admissibility. The work introduces energy-preserving DMV formulations, extends maximality results to these settings, and suggests a conjecture linking strong convergence to maximal computable DMV solutions and oscillatory behavior to nonexistence of a maximal computable DMV limit. Altogether, the results illuminate when entropy-based selection principles are consistent with measure-valued limits and have implications for numerical design and interpretation of Euler-system approximations.

Abstract

We show that the measure-valued solutions of the Euler system of gas dynamics generated by oscillatory sequences of consistent approximations violate the principle of maximal entropy production formulated by Dafermos. Numerical results illustrate that solutions obtained by standard numerical methods may be oscillatory and thus do not comply with the Dafermos criterion.
Paper Structure (19 sections, 9 theorems, 124 equations, 4 figures, 1 table)

This paper contains 19 sections, 9 theorems, 124 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Euler system of gas dynamics
  3. Dafermos' admissibility criterion
  4. Dafermos criterion in the class of measure--valued solutions
  5. Measure--valued solutions with small energy defect
  6. Energy preserving measure--valued solutions
  7. Dissipative measure--valued (DMV) solutions
  8. DMV solutions
  9. Basic properties of DMV solutions
  10. Main result on regularity of maximal DMV solutions
  11. Consistent approximations, main result on convergence
  12. Proof of Theorem \ref{['dT1']}
  13. Concatenation property for DMV solutions
  14. Vanishing energy defect for admissible solutions
  15. Proof of Theorem \ref{['cadT']}
  16. ...and 4 more sections

Key Result

Proposition 2.2

Let $\Omega \subset R^d$, $d=2,3$ be a bounded domain with smooth boundary. Suppose the thermodynamic functions $p$, $e$, and $s$ satisfy hypotheses r1a--r3a. Let the initial data belong to the class where $\underline{s} \in R$ is a given constant. Then the Euler system E1--E3, E4 admits a DMV solution in $(0,T) \times \Omega$ specified in Definition Dd2. In addition, the solution satisfies

Figures (4)

  • Figure 1: Density computed by the first and higher-order VFV methods at time $T=2.0$.
  • Figure 2: Cesàro averages of the density computed by the first and higher-order VFV methods at time $T=2.0.$
  • Figure 3: Time evolution of the $L^1$-norm of the Reynolds defect and of the energy defect, $t \in [0,2]$.
  • Figure 4: Time evolution of the total entropy $\mathcal{S}(t)$ (left) and the entropy defect $D_{\rm Ent}(t)$(right), $t \in [0,2]$.

Theorems & Definitions (23)

  • Definition 2.1: DMV solution
  • Proposition 2.2: Global existence
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1: Maximal DMV solution
  • Theorem 3.2: Regularity of maximal DMV solutions
  • Corollary 3.3
  • Definition 4.1: Consistent approximation
  • Proposition 4.2
  • Definition 4.3: Oscillatory consistent approximations
  • ...and 13 more