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

Heiko Gimperlein

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

T0 review · 0 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper shows that for piecewise constant fan subsolutions, action-rate and entropy-rate admissibility coefficients both reduce to a single kinetic/internal-energy decomposition, and that for a specific contact-discontinuity Riemann fan

desk verdict A clean, honest computation: the K/I decomposition is new and the certified KS-fan evaluation is convincing, with the main caveats being the cited convex-integration identities and the unexecuted exact-arithmetic certificate. read the letter →

arxiv 2608.02482 v2 pith:JAVYK26T submitted 2026-08-03 math.AP

Heiko Gimperlein This is my paper · ORCID

classification math.AP MSC 35Q3135L6576N10

keywords isentropicEulerequationsconvexintegrationfansubsolutionentropyratecriterionleastactionprinciplecontactdiscontinuityadmissibilitycriteriaRiemannproblem

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's central claim is that two rival selection criteria for weak solutions of the isentropic Euler equations—the entropy rate criterion and the action rate criterion—are governed by the same pair of numbers whenever the solutions come from a piecewise constant fan subsolution. For such fans the paper derives explicit formulas and proves the reduction D = K/2 + I and E = -K/2 + I, where K is a weighted kinetic-energy mismatch and I a weighted internal-energy mismatch; the criteria therefore agree exactly when |I| < |K|/2. The paper then applies this to a fan constructed numerically for Riemann data whose classical solution is a planar contact discontinuity. Certified exact rational computations show D > 0 and E < 0 at the exact fan, so both criteria prefer every convex integration realization generated by the fan to the classical contact discontinuity. This matters because it says the standard selection criteria do not restore uniqueness for this Riemann problem; they point instead toward the nonunique multidimensional solutions.

What carries the argument

The central object is a piecewise constant fan subsolution: a self-similar partition of spacetime into conical regions on which the coarse density, velocity, and relaxed stress are constant and satisfy a strict convex-integration subsolution condition; each convex integration realization has density equal to the fan density and |v|^2 equal to a prescribed constant in each intermediate region. The load-bearing identities are D = K/2 + I and E = -K/2 + I, where K represents the weighted kinetic-energy mismatch and I the weighted internal-energy mismatch. These reduce the comparison of the two admissibility criteria to the location of the pair (K, I) relative to the lines I = ±K/2. The paper al

What would settle it

Evaluate D and E at the exact fan parameters with an independent exact-arithmetic implementation: if either D is not positive or E is not negative, the central conclusion fails. Because the certified margins are large (D > 6609, E < -1407, correction bound 2.45e-13), a counterexample would have to come from a certified error in the auxiliary verification or from a violation of the |v|^2 = C_i identity on a set of positive measure.

Watch

Extended reading notes

Core claim

The core discovery is that all convex integration realizations belonging to the same admissible piecewise constant fan subsolution are identical from the standpoint of both criteria: their action difference and energy-rate difference relative to any common piecewise constant self-similar reference solution depend only on the fan, not on the internal oscillations inserted by convex integration. The action and entropy coefficients are computed in closed form as sums over the overlap of fan and reference regions, and are decomposed into kinetic and internal mismatches. For the specific planar-contact-discontinuity fan, exact arithmetic shows the action coefficient is positive and the entropy co

Load-bearing premise

The sign transfer from rational approximations to the exact fan assumes the Newton–Kantorovich certificates are correct and that every convex integration realization of the fan satisfies density equal to the fan density and |v|^2 equal to the prescribed constant almost everywhere in each fan region; if the constructed solutions only approximate these identities, the signs of the coefficients could change.

Editorial extensions

If this is right

Any two convex integration realizations generated by the same fan are indistinguishable to both criteria: selection, if it happens, is determined by the fan, not by the realization.
The two criteria agree precisely when the weighted kinetic mismatch dominates the internal mismatch; outside that sector they select opposite solutions.
For the contact-discontinuity Riemann data, neither the entropy rate criterion nor the action criterion selects the classical solution—both select the convex integration family.
The explicit finite-dimensional formulas turn the admissibility comparison into a checkable algebraic condition, so the same computation can be rerun for any other fan subsolution.

Reading between the lines

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

If the K/I reduction extends beyond piecewise constant fans to more general subsolutions, the (K, I) plane could become a general diagnostic for when dissipation-based selection criteria agree; this extension is not proved in the paper.
The result suggests that scalar dissipation or action rates may be structurally blind to the difference between a one-dimensional contact and genuinely multidimensional oscillations, since both carry the same macroscopic energy profile; that reading goes beyond the paper.
The certification pipeline—exact rational coefficient evaluation plus interval/Newton–Kantorovich transfer—could be applied to other numerically constructed fans, for example the two-shock fans where the paper predicts the criteria disagree, to test the characterization.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Pairwise, this paper earns its claims. The new content is the finite-dimensional reduction of the two admissibility criteria for piecewise constant fan subsolutions—the formulas D = 1/2 K + I and E = -1/2 K + I, the phase diagram, and the explicit certified evaluation showing that for the Krupa–Székelyhidi fan both the entropy-rate and least-action criteria prefer the convex-integration family to the planar contact discontinuity. That is a concrete, reusable result, and it is not fitted: the fan parameters come from [11], and the sign inequalities are strict with enormous margins.

The paper does well in being precise about what it does not prove. Section 6 explicitly separates the conditional transfer to the exact fan from the existence argument in [11], and Section 7 describes the exact-arithmetic certification in enough detail that an independent reader can rerun it. The Maple worksheet and certify.py scripts are readily available. That is the right way to handle computer-assisted parts.

Soft spots, in proportion. The strongest claim—that every convex-integration realization generated by the KS fan is preferred to the classical solution—rests on the pointwise identities ρ = ρ̄ and |v|² = C_i a.e., which are imported from [11] and [13] and not re-derived here. That is a legitimate citation, not a circular step, but it makes the 'every realization' wording conditional on the cited construction being exactly what the author assumes. I would prefer the author to flag that dependency more prominently. The Newton–Kantorovich transfer itself is the other soft spot: I did not execute certify.py, so I am relying on the description plus the large margins. The margins are so large that a bug capable of flipping the sign is implausible, but independent execution is the natural check. The specially constructed pressure law is artificial, which the author acknowledges in the introduction; it is standard for this nonuniqueness program.

On balance, the central argument holds up. The decomposition theorem is elementary and correct; the application is new and convincing. This paper is for people working on selection criteria and nonuniqueness for the multidimensional Euler equations, and it deserves a serious referee. I would send it to peer review with minor revisions.

Referee Report

0 major / 4 minor

Summary. The paper studies admissibility criteria for weak solutions of the two-dimensional isentropic Euler equations obtained by convex integration from piecewise constant fan subsolutions. The main structural result is that, relative to any piecewise constant self-similar reference solution, the action-rate and entropy-rate coefficients of every convex integration realization generated by a given fan depend only on the fan, not on the particular realization. These coefficients are shown to decompose as D = K/2 + I and E = -K/2 + I in terms of a weighted kinetic-energy mismatch K and internal-energy mismatch I, yielding a complete two-sector characterization of when the action and entropy rate criteria agree. The theory is applied to the Krupa–Szekelyhidi fan for Riemann data whose classical solution is a planar contact discontinuity. Using exact rational arithmetic and certified Newton–Kantorovich bounds, the paper establishes D_KS > 0 and E_KS < 0 for the exact fan, so that both criteria strictly prefer every convex integration solution associated with that fan to the classical contact solution.

Significance. If the result holds, it is a valuable and concrete contribution to the ongoing discussion of selection criteria for nonunique weak solutions of the compressible Euler equations. The paper provides a transparent, essentially algebraic reduction of the two criteria to a pair of coefficients, and it demonstrates on a nontrivial example that both criteria can select the same 'wild' solutions over the classical self-similar solution. The strengths of the paper are its explicit, parameter-free derivation of the D/E decomposition; the clean separation of the finite-dimensional coefficient evaluation from the existence of the exact fan; and the use of exact rational arithmetic and certified bounds with enormous margins rather than floating-point heuristics. The two verification concerns raised in the stress-test — the exact pointwise identities for convex integration realizations and the unexecuted certification scripts — are real but, on reading the manuscript, they do not land as demonstrated errors: the identities are stated as explicit hypotheses imported from the cited literature, and the certificate is described in enough algorithmic detail with code made available. I did not indepen

minor comments (4)

[Section 2, items (i)-(iii)] The pointwise identities ρ = ρ̄ and |v|^2 = C_i a.e. in each fan region are load-bearing for Proposition 2.8 and hence for the entire paper. Since they are imported from the convex integration literature, please give the specific theorem or proposition number in [11] or [13] that guarantees them, rather than only the summary statement, so that a reader does not have to reconstruct the argument.
[Sections 6.2 and 7 / Remark 7.4] The proof of Proposition 6.3 and 6.4 assumes e(ρ_3^*) = ê_3, which the exact normalization of [11] does not satisfy; the corrected chain rule appears only in Remark 7.4. Please integrate the affine-e computation into the main proof of the Lipschitz bounds and state Theorem 7.3 under the true normalization, so that the formal proof does not rest on a provisional assumption.
[Section 7 and Theorem 7.2] The Newton–Kantorovich certificates for δ_KS, L_D, L_E, and the persistence of the inequality constraints are delegated to certify.py and the Maple worksheet. For archival and reproducibility, include the scripts or a detailed verification log as supplementary material, and state explicitly in the main text that the output has been generated by running them to completion.
[Theorem 6.5] The statement that the margin is 'greater than 10^10' is informal. Please replace it with the formal certified inequalities, e.g. L_D δ_0 ≤ 1.4×10^{-8} < 6609 and L_E δ_0 ≤ 1.3×10^{-8} < 1407, which are the actual quantitative bounds.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the rate-coefficient formulas are algebraic identities and the KS evaluation is external, not fitted.

full rationale

The derivation chain is not circular. The structural formulas (3.3), (3.4), Proposition 5.1, and Theorem 4.1 are exact algebraic consequences of the definitions of H, L, K, I and the stated fan assumptions, with no fitted parameters. The values D_KS and E_KS are evaluated at the rational parameter vector taken from the external Krupa-Szekelyhidi construction [11], not chosen to force the desired signs; the margins are explicit (D>6609, E<-1407). The passage to the exact fan (Theorems 6.5, 7.2, 7.3) is a standard Newton-Kantorovich argument with certified Lipschitz bounds and correction estimates, conditional on externally stated residuals and Jacobian data from [11,12]. The author's own prior work [9,10] supplies only the definition of the LAAP0 criterion; the comparison is then evaluated, not assumed. Some verification material is not reproduced in the manuscript ('We do not reproduce the existence argument', Section 6; certify.py is ancillary), but this is an omitted-detail / verification-level gap, not an input-output identification. No step reduces a claimed prediction to a fitted parameter or to a self-citation, so no significant circularity is present.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The structural theorem has no free parameters. The application uses the externally computed KS fan parameters from [11] as inputs; these are not fitted in the present paper. The paper relies on convex-integration existence theory, the Newton–Kantorovich theorem, and the specific thermodynamic normalization of [11,12].

free parameters (1)

KS fan parameters (rho_i, C_i, e_i, nu_-, nu_1, nu_2, nu_+, v_±, s, rho_0) = Exact rational values in Table 1 (from [11])
External inputs, not fitted in this paper. The application's sign conclusions D_KS > 0 and E_KS < 0 are evaluated at these values; the structural theorem is independent of them.

assumptions (5)

domain assumption For every admissible fan subsolution, convex integration produces exact entropic weak solutions with rho = bar-rho and |v|^2 = C_i a.e. on each fan region.
Invoked in Section 2 immediately after Definition 2.6 and used in Proposition 2.8 to show H and L depend only on the fan. Cited from [2,3,11], not proved here.
domain assumption The exact Krupa–Szekelyhidi fan subsolution exists and is strictly admissible for the contact Riemann data.
Taken from [11]. Section 7 transfers sign inequalities to this exact fan but does not prove its existence.
standard math Newton–Kantorovich theorem for the certified correction-size bound.
Used in Theorem 7.2 to convert residual, Jacobian lower bound, and Lipschitz bound into existence, uniqueness, and the estimate delta_KS <= 2.45e-13.
domain assumption The thermodynamic normalization e''(rho_3) = 0 and affine interpolation near rho_3 used in [11,12].
Needed in Section 6.2 and Remark 7.4 to evaluate e(rho_3*) and the derivatives of D_KS and E_KS with respect to rho_3. Remark 7.4 gives adjusted constants if this normalization is only approximate.
domain assumption The fan and reference profiles agree on the exterior regions and share the same Riemann data.
Used in Proposition 3.3 and the formulas of Proposition 5.1; if the exterior states differed, the integrals E and D would contain boundary terms.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Admissibility criteria for convex integration fan solutions and contact discontinuities in the Euler equation." pith.science (2026). https://pith.science/paper/JAVYK26T

@misc{pith2026260802482,
  author       = {Pith},
  title        = {Pith review of: Admissibility criteria for convex integration fan solutions and contact discontinuities in the Euler equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JAVYK26T}},
  note         = {Machine review of arXiv:2608.02482}
}

read the original abstract

For piecewise constant fan subsolutions to the isentropic Euler equations, the entropy and action rates of any associated convex integration solution, relative to a common reference solution, depend only on the fan and are computed explicitly. Moreover, these explicit expressions may be decomposed into a kinetic-energy mismatch and an internal-energy mismatch from which we characterize agreement of Dafermos' entropy rate criterion with action rate criteria. The known disagreement for the Chiodaroli-Kreml two-shock data is recovered. For the contact-discontinuity solutions of Krupa and Sz\'{e}kelyhidi, certified exact-arithmetic computations show that both criteria prefer the convex integration solutions to the classical contact discontinuity. The same agreement holds, by a purely analytic argument, for Horimoto's contact-discontinuity solutions with arbitrary strictly increasing pressure laws.

Figures

Figures reproduced from arXiv: 2608.02482 by the authors.

**Figure 1.** Selection regions in the (K, I)-plane. Action and entropy criteria agree in the left and right sectors. Point KS shows the location of the Krupa–Sz´ekelyhidi solution [11]. 2) The approach is applied to the numerically constructed solutions from [11]. We show that both admissibility criteria prefer the convex integration solutions to the classical contact discontinuity. Ancillary codes rigorously verify these numeri… view at source ↗

Discussion (0). Continue with ORCID to comment.

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.