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Maximal entropy production principle and the Euler system of gas dynamics

T0 review · 2 major / 2 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read The maximal entropy production principle excludes the standard self-similar solution for certain two-dimensional Riemann problems in the Euler system.

desk verdict The paper gives a direct 2D Riemann counterexample showing the self-similar solution does not maximize entropy production, plus a general construction for arbitrary non-decreasing entropy profiles. read the letter →

arxiv 2605.26687 v1 pith:JR7I2M4V submitted 2026-05-26 math.AP

Elisabetta Chiodaroli , Eduard Feireisl , Ondřej Kreml , Simon Markfelder This is my paper · ORCID
classification math.AP
keywords EulerequationsentropyproductionweaksolutionsRiemannproblemconvexintegrationgasdynamicsmaximal
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tests whether the maximal entropy production principle can select physically relevant weak solutions to the Euler equations when the entropy inequality alone fails to do so. For a specific family of two-dimensional Riemann initial data the usual self-similar solution does not maximize the entropy production rate. The authors therefore conclude that the principle eliminates the solution that appears physically natural. They also construct, for a wide class of initial data, entropy-admissible weak solutions whose total entropy can follow any prescribed non-decreasing function of time.

What carries the argument

The maximal entropy production principle, which selects among entropy-admissible weak solutions the one that maximizes the instantaneous rate of entropy increase.

What would settle it

A direct numerical or experimental measurement, for the specific Riemann data of the example, of whether the observed entropy production rate equals the rate of the self-similar solution or exceeds it.

Watch

Extended reading notes

Core claim

For Riemann initial data in two space dimensions the standard self-similar solution fails to attain the maximal possible rate of entropy production among entropy-admissible weak solutions. Hence the maximal entropy production principle rules out this solution. In addition, for a large class of initial data there exist entropy-admissible weak solutions whose integrated entropy production can be prescribed as any non-decreasing function of time.

Load-bearing premise

The maximal entropy production principle is a physically meaningful selection criterion that should recover the intuitively relevant self-similar solution for the given Riemann data.

Editorial extensions

If this is right

  • The self-similar solution is excluded by the maximal entropy production principle for the constructed two-dimensional Riemann data.
  • Entropy-admissible weak solutions with arbitrarily prescribed non-decreasing total entropy profiles exist for a broad class of initial data.
  • The non-uniqueness of weak solutions persists even after imposing the maximal entropy production criterion.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Alternative selection mechanisms beyond maximal entropy production may be required to isolate physically observed solutions.
  • The freedom to choose the entropy profile suggests that dissipative effects in the model can be tuned independently of the initial data.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper uses convex integration to address selection of weak solutions to the 2D Euler system of gas dynamics under the entropy inequality. For specific Riemann initial data it constructs an explicit counter-example showing that the standard 1-D self-similar solution does not attain the supremum of the entropy production rate. It further proves that, for a large class of initial data, entropy-admissible weak solutions exist whose total entropy profile can be any prescribed non-decreasing function.

Significance. If the constructions are correct, the work supplies a mathematically rigorous negative answer to whether the maximal entropy production principle recovers the intuitively physical self-similar solution, and demonstrates that the admissible entropy profiles are essentially arbitrary. The explicit counter-example and the flexible profile construction are concrete contributions to the literature on ill-posedness and selection criteria for hyperbolic systems.

major comments (2)
  1. [§3] §3 (counter-example construction): the verification that the self-similar solution fails to maximize the entropy production rate relies on an explicit comparison of total entropy production; the argument would be strengthened by an explicit formula or numerical value for the production rate attained by the self-similar solution versus the constructed competitor.
  2. [Theorem 1.2] Theorem 1.2 (arbitrary entropy profiles): the statement that any non-decreasing profile is attainable is load-bearing for the second claim; the proof sketch in §4 should clarify how the convex-integration scheme controls the integrated entropy production while preserving the entropy inequality pointwise.
minor comments (2)
  1. [§2] The notation for the entropy production rate functional is introduced without a numbered equation; adding an explicit definition (e.g., Eq. (2.3)) would improve readability.
  2. [Figure 1] Figure 1 (Riemann data diagram) lacks axis labels and a caption explaining the wave configuration; this is a minor clarity issue.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the work, and the recommendation for minor revision. We address each major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (counter-example construction): the verification that the self-similar solution fails to maximize the entropy production rate relies on an explicit comparison of total entropy production; the argument would be strengthened by an explicit formula or numerical value for the production rate attained by the self-similar solution versus the constructed competitor.

    Authors: We thank the referee for this constructive suggestion. The comparison in §3 already establishes that the convex-integration competitor attains a strictly higher total entropy production rate than the self-similar solution, but we agree that an explicit formula or numerical illustration would make the distinction more transparent. In the revised manuscript we will add the explicit entropy-production formula for the self-similar Riemann solution together with a direct numerical comparison against the rate realized by the constructed weak solution. revision: yes

  2. Referee: [Theorem 1.2] Theorem 1.2 (arbitrary entropy profiles): the statement that any non-decreasing profile is attainable is load-bearing for the second claim; the proof sketch in §4 should clarify how the convex-integration scheme controls the integrated entropy production while preserving the entropy inequality pointwise.

    Authors: We agree that additional detail on this point will strengthen the exposition. In the revised version we will expand the proof sketch in §4 to explain precisely how the convex-integration iteration is modified so that the integrated entropy production can be prescribed arbitrarily (subject only to monotonicity) while the entropy inequality is enforced pointwise at every stage of the construction. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper establishes mathematical counterexamples and existence results for the Euler system via convex integration. The key statements—that the 1-D self-similar solution fails to maximize entropy production for given 2-D Riemann data, and that arbitrary non-decreasing entropy profiles are attainable—are direct constructions from the method, not reductions to fitted inputs, self-definitions, or load-bearing self-citations. No derivation step equates a claimed prediction or uniqueness result to its own inputs by construction. The work is self-contained against external mathematical benchmarks.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Only the abstract is available; no explicit free parameters, invented entities, or non-standard axioms are identifiable from the given text.

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Cite this review

Pith. "Pith review of Maximal entropy production principle and the Euler system of gas dynamics." pith.science (2026). https://pith.science/paper/JR7I2M4V

@misc{pith2026260526687,
  author       = {Pith},
  title        = {Pith review of: Maximal entropy production principle and the Euler system of gas dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JR7I2M4V}},
  note         = {Machine review of arXiv:2605.26687}
}
read the original abstract

Convex integration has revealed that the Euler system of gas dynamics is ill-posed in the class of weak solutions even if the entropy inequality is imposed as an additional constraint. A natural question arises, namely, if a physically relevant solution can be selected by maximizing the entropy production rate. Firstly, we present an example of Riemann initial data in 2-D, for which the standard self-similar solution fails to satisfy the maximal entropy production principle. Hence, maximizing the entropy production rate rules out the 1-D self-similar solution which intuitively seems to be the physically relevant solution in this context. Secondly, we show for a large class of initial data that there exist entropy admissible weak solutions with an arbitrary (non-decreasing) total entropy profile.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Admissibility criteria for convex integration fan solutions and contact discontinuities in the Euler equation

    math.AP 2026-08 accept novelty 6.0 of 10

    For piecewise constant fan subsolutions of the isentropic Euler equations, the entropy-rate and action-rate admissibility criteria reduce to two coefficients, and for the Krupa–Szekelyhidi contact-discontinuity exampl...

Reference graph

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