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On "discrete" solutions of the Euler system of gas dynamics

T0 review · 4 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read For a dense set of initial density and entropy data, the Euler system admits infinitely many discrete, entropy-admissible weak solutions with increasing total entropy, even converging to a prescribed terminal entropy profile in the monoatom

desk verdict A well-put convex-integration construction for discrete full-Euler solutions that hinges on an unverified imported proposition; referee it, but make the authors check the black box. read the letter →

arxiv 2608.00126 v1 pith:TCWU6GBG submitted 2026-07-31 math.AP

Anna Abbatiello , Eduard Feireisl This is my paper · ORCID
classification math.AP MSC 35Q3135D3035L6576N10
keywords Eulersystemofgasdynamicsdiscretesolutionsconvexintegrationentropyadmissibilityill-posednesslong-timebehaviourweakterminalprofile
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 sets out to show that the entropy inequality is almost powerless as a selection criterion for the full Euler system of gas dynamics. It constructs, for a dense set of initial densities and entropies, an initial momentum such that infinitely many weak solutions coexist. These solutions are discrete, meaning each one attains only finitely many constant states, so the non-uniqueness is not merely an artifact of wild oscillations. The total entropy can be made to increase from one solution to the next, and in the three-dimensional monoatomic case the entropy can also be steered toward any prescribed Riemann-integrable terminal profile as time approaches the horizon. If the argument holds, physically relevant admissibility criteria based on entropy growth alone cannot single out a unique solution.

What carries the argument

The load-bearing object is an imported convex-integration lemma (Proposition 4.1) asserting that on each subdomain of a finite partition, the overdetermined local problem (3.4)–(3.6)—weakly divergence-free momentum with test functions not vanishing on the boundary, plus the pointwise kinetic-energy constraint |m_n|^2/(2ϱ0,n)=Λ−(d/2)p(ϱ0,n,s0,n)—admits infinitely many solutions taking values in a finite set of vectors on a sphere. Around this, the paper assembles global solutions by time-splicing: concatenating solutions on consecutive intervals while augmenting entropy at each junction and adjusting Λ to keep total energy continuous; for terminal profiles, it uses a nested refinement of subd

What would settle it

The decisive check is to test the local construction on a single measurable subdomain with no boundary regularity and with test functions that do not vanish on the boundary: exhibit one such subdomain for which the overdetermined system (3.4)–(3.6) has no bounded finite-valued momentum field. If such a subdomain exists, the imported lemma fails at the level of generality used and the dense-set theorems collapse.

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Extended reading notes

Core claim

The central claim is Theorem 2.3 together with Theorem 2.9: there is a set of initial pairs (density, entropy), dense in the L^q topology, such that for each pair one can choose a bounded initial momentum for which the complete Euler system has infinitely many admissible weak solutions, indexed by λ≥0. Each solution takes values in finitely many constant states (ϱ_k, m_k, s_k). The solutions are ordered by total entropy: for λ1<λ2, the integral of ϱλ1 sλ1 is pointwise no larger than that of ϱλ2 sλ2, and they differ on a set of positive measure. In the monoatomic case d=3, γ=5/3, the same initial data also admit solutions whose entropy converges strongly in L^q to an arbitrary Riemann-integra

Load-bearing premise

The entire construction rests on an imported convex-integration proposition that is assumed to supply infinitely many finite-valued momenta solving the local problem (3.4)–(3.6) on arbitrary measurable subdomains with no boundary regularity and non-vanishing boundary test functions; the paper does not verify the hypotheses of that proposition in this setting.

Editorial extensions

If this is right

  • If the theorem is correct, the entropy inequality alone cannot select a unique weak solution: a dense set of initial density/entropy profiles is accompanied by infinitely many entropy-admissible solutions.
  • The constructed solutions are discrete, so non-uniqueness survives even when solutions are required to attain only finitely many constant states, not just among highly oscillatory fields.
  • The family of solutions is ordered by strictly increasing total entropy, so none of them is maximal with respect to the standard entropy-rate orderings considered in the literature; those criteria therefore do not, by themselves, eliminate such solutions.
  • In the 3D monoatomic case, the same initial data produce solutions with any prescribed Riemann-integrable terminal entropy profile, and the required initial momentum does not depend on the profile; entropy can thus be made to approach equilibrium at arbitrarily fast or slow rates.
  • The same machinery yields time-periodic entropy-admissible solutions and solutions whose kinetic energy decays to zero along a prescribed envelope, further illustrating the flexibility of weak solutions.

Reading between the lines

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

  • The paper leaves implicit that if the imported convex-integration lemma were shown to hold on all measurable subdomains without boundary regularity, the same time-splicing argument would likely produce discrete wild solutions for a broader class of equations of state, including the radiation-pressure models mentioned in the text.
  • A natural testable extension is to ask whether the discrete solutions can be forced to satisfy additional admissibility conditions, such as Clausius–Duhem-type inequalities or maximal entropy production; the paper's own discussion suggests that maximal entropy production would remove its solutions, but this is not proved.
  • Because the initial momentum can be chosen independently of the terminal entropy profile, the result suggests that the reachable set of entropy profiles from a fixed initial state is extremely large; one could try to determine whether any monotone path of entropy profiles can be realized in the same way.
  • In numerical terms, the terminal-entropy theorem provides a benchmark: any numerical scheme that provably converges to a unique entropy-admissible solution for these data would contradict the paper's conclusion, so the theorem delineates what numerical entropy dissipation can and cannot guarantee.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. The paper constructs "discrete" entropy-admissible weak solutions to the full compressible Euler system (1.1)-(1.8) in bounded domains Ω⊂R^d, d=2,3, for polytropic gases with γ≤5/3. The authors fix a finite partition Ω=∪Ω_n into arbitrary measurable sets and piecewise-constant initial density and entropy. On each cell they solve a local 'incompressible-type' problem (3.4)-(3.6) for the momentum using convex integration imported from Luo-Xie-Xin [17], then piece the cells together. Because the local momentum takes values in a finite set of vectors and the densities/entropies are constant, the resulting weak solutions attain only finitely many constant states. By concatenating such solutions on consecutive time intervals and increasing the entropy at the interfaces while preserving the total energy through condition (5.4)-(5.5), they obtain infinitely many admissible solutions from the same initial data with strictly increasing total entropy (Theorem 2.3), and similar results with prescribed initial kinetic energy (Theorem 2.6) or total energy (Corollary 2.7). In the special case d=3, γ=5/3, an infinite concatenation with refining spatial partitions yields solutions whose entropy converges to an arbitrary prescribed Riemann-integrable profile at the terminal time (Theorem 2.9). The final sections discuss consequences for maximality criteria and long-time behaviour, including time-periodic variants and convergence to equilibria.

Significance. If correct, the paper would considerably strengthen the known non-uniqueness picture for the complete Euler system: the entropy inequality would impose almost no restriction on the structure or long-time behaviour of weak solutions, and the initial kinetic energy need not be large. The construction is conceptually clean and the entropy-concatenation mechanism is transparent. The paper also explicitly acknowledges that the resulting solutions are not maximal in DiPerna's or Dafermos' sense, which is an honest and useful observation. The main barrier to accepting the result is that the sole non-elementary ingredient, Proposition 4.1, is imported without a verification of its hypotheses; the paper's entire edifice rests on that black box.

major comments (4)
  1. [Section 4, Proposition 4.1 and Remark 4.2] Proposition 4.1 is the only non-elementary input, but its hypotheses are not checked for the present setting. The subdomains Ω_n in (3.1) are arbitrary measurable sets, no regularity of ∂Ω_n is assumed, and the test functions in (3.4)–(3.5) are C^1([0,T]×Ω_n) and do not vanish on ∂Ω_n. In addition, (3.9) asserts that the solutions satisfy m_n∈C_weak([0,T];L^q) and m_n(0)=m_n(T)=0; this endpoint condition is essential for the concatenation in Section 5. The paper only identifies parameters in Remark 4.2; it does not state the exact hypotheses of [17, Prop. 1] nor verify that they hold for arbitrary measurable partitions and non-compact test functions. If [17] requires, say, a smooth or periodic domain, or test functions with zero boundary values, then the local problem (3.4)–(3.6) may have no solution and the derivation of (4.5)–(4.9), Proposition 4.3, and all main theorems collapses. Ple
  2. [Section 4, paragraph after (4.9) and Proposition 4.3] The passage from a solution with m(0)=m(T)=0 to an admissible weak solution with a prescribed nonzero initial momentum m0 is described as 'a simple time shift,' but the construction is not given. The weak formulation requires the initial datum m0 in (2.5); if the constructed trajectory has zero trace at t=0, the initial momentum cannot be arbitrary. Please explain explicitly how m0 is selected (e.g., as a trace m(τ) for a suitable τ), prove that the shifted solution satisfies Definition 2.1, and ensure that the energy balance (2.6) holds at the initial time.
  3. [Section 4, Eq. (4.1)–(4.3)] The number of constant vectors is inconsistent: the text and Eqs. (4.1)–(4.2) contain 'd(d+3)/d' while Proposition 4.1 in (4.3) reads 'd(d+3)/2'. This is not merely cosmetic, since the cardinality of the finite set in (4.3) is used in the discreteness statement. Please correct the count and confirm that it matches [17, Lemma 3].
  4. [Section 5.1, alternatives after Eq. (5.5)] In the proof of Theorems 2.3/2.6, the boundary case 1 - d/2(γ-1)=0 (i.e., d=2,γ=2, which is allowed by the assumptions) is not covered. Alternative 1 is stated only for d=3,γ=5/3, and alternative 2 with λ>0 requires the coefficient to be positive. For d=2,γ=2, the same argument as in alternative 1 (bΛ=Λ, bs=s+λ) applies. Please extend alternative 1 to γ=1+2/d, or otherwise handle this case.
minor comments (3)
  1. [Section 4, first paragraph] The phrase 'there exist d(d+3) d vectors' is garbled and should be corrected to 'd(d+3)/2 vectors' (or whatever the correct count is).
  2. [Eq. (4.2)] In the displayed convex-hull condition, the kinetic term appears as '1/d |m^ℓ_n|/ϱ_{0,n}' but should be '1/d |m^ℓ_n|^2/ϱ_{0,n}', consistent with (3.5).
  3. [Section 5.2, proof of Theorem 2.9] The sentence 'the density remains continuous' is inaccurate; the density is piecewise constant on the original partition. Also, in Remark 2.11, '˜ϱ' should presumably be 'es'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the entropy-ordering and terminal-entropy conclusions are constructive consequences of an external convex-integration input, not restatements of the hypotheses.

full rationale

I found no step in which a claimed prediction reduces by construction to an input, and no fitted parameter is renamed as a prediction. The paper reduces the full Euler system in Section 3 to the local problem (3.4)–(3.6) on an arbitrary measurable partition, and the existence of the momentum fields m_n is imported from an external theorem, Luo–Xie–Xin [17, Proposition 1], quoted as Proposition 4.1. This is not a self-citation, and the proof is conditional on it; whether the hypotheses of [17] really cover the present boundary-free test functions and arbitrary measurable subdomains is a correctness gap, not circularity. The increasing-entropy property (2.9) is not assumed: in Section 5.1 the authors explicitly choose augmented entropies bs0,n > s0,n and a constant bΛ satisfying the energy-matching relation (5.4)–(5.5), with the two stated alternatives for polytropic gases, so the entropy ordering is a derived consequence of the concatenation construction. Similarly, Theorem 2.9 is constructive: the entropy at each concatenation time is defined as the infimum of the prescribed terminal profile es over a nested, refining partition, and the L1 convergence follows from Darboux-sum convergence and (5.8); the terminal profile is a target used to define the sequence, not a conclusion hidden in the hypotheses. The self-citations that appear are not load-bearing: [13] is cited for the reformulation strategy, while the key existence input is the external [17]; and [15] is cited only to interpret Dafermos-maximality, with the non-maximality of the constructed solutions demonstrated in-text by choosing an earlier first concatenation time. The central claims therefore have independent mathematical content beyond their inputs.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

No new physical entities, forces, or dimensions are introduced. The free parameters Lambda, entropy jumps, time partitions, and domain refinements are auxiliary choices in the construction, not fitted parameters. The main external input is the convex-integration proposition of Luo-Xie-Xin, whose applicability to irregular subdomains is not verified.

free parameters (4)
  • Lambda = positive constant constrained by Lambda > (d/2)p(rho0,n,s0,n) and, in Theorem 2.9, Lambda > rho0 e(rho0,e s)
    Sets the kinetic-energy level in (3.6); all constructed solutions share this total-energy parameter and it must be chosen by hand to satisfy positivity and energy-conservation constraints.
  • Entropy jump values bs0,n (lambda) = nonnegative increments satisfying (5.4)-(5.5)
    At each concatenation time the entropy is raised to a new piecewise-constant profile; the increment is a free choice subject to bLambda - (d/2)p > 0 and energy conservation.
  • Time partition {tau_m} = arbitrary increasing sequence; tau1 from the full-measure set in Proposition 4.3
    Determines at which times entropy jumps occur; different partitions produce different solutions and control the rate of entropy production or terminal convergence.
  • Domain refinements Omega_{m,n} = nested partitions with diam -> 0
    Used in Theorem 2.9 to approximate the Riemann-integrable terminal entropy profile; the diameters are chosen freely.
assumptions (6)
  • domain assumption Polytropic equation of state p = (gamma-1)rho e with 1 < gamma <= 5/3 (and gamma = 5/3 in Theorem 2.9)
    The energy-conservation relation (5.5) and the interval for E0 in Corollary 2.7 depend on this constitutive law; the terminal-entropy theorem only treats the monoatomic case.
  • domain assumption Gibbs relation (1.4) implies e(rho,s) is strictly increasing in s
    Used to solve (5.5) for bs0,n(lambda) and to ensure the entropy ordering is compatible with energy conservation.
  • standard math Luo-Xie-Xin convex integration proposition [17, Proposition 1] supplies infinitely many m_n solving (3.4)-(3.5) with kinetic-energy constraint (3.6)
    The paper does not reprove this result; the entire discrete-solution construction reduces to applying this external proposition on each Omega_n.
  • standard math Piecewise-constant functions are dense in L^q and nested refinements with diameter -> 0 approximate Riemann-integrable profiles in L^1
    Used to turn the dense set I_{rho,s} of piecewise-constant data and the terminal entropy profile into L^q statements.
  • ad hoc to paper The subdomains Omega_n in the decomposition (3.1) may be arbitrary measurable sets, and Proposition 4.1 applies with test functions that do not vanish on their boundaries
    The authors assert this without proof; it is not obviously a hypothesis of [17, Proposition 1] and is the weakest assumption of the paper.
  • domain assumption Bounded domain Omega, impermeable boundary m.n = 0, and no vacuum zones
    These are the hypotheses of Definition 2.1; the boundary condition is understood weakly as described in Remark 2.2.

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Pith. "Pith review of On "discrete" solutions of the Euler system of gas dynamics." pith.science (2026). https://pith.science/paper/TCWU6GBG

@misc{pith2026260800126,
  author       = {Pith},
  title        = {Pith review of: On "discrete" solutions of the Euler system of gas dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TCWU6GBG}},
  note         = {Machine review of arXiv:2608.00126}
}
read the original abstract

The method of Convex Integration has revealed a number of rather disturbing facts concerning well-posedness of the Euler system of gas dynamics. In particular, there is a dense set of "wild" initial data, for which the problem admits infinitely many physically admissible (entropy) weak solutions. We identify the class of initial data enjoying the following properties: (a) they give rise to a family of weak solutions with increasing entropy profiles; (b) the solutions are "discrete", meaning they attain only a finite number of constant states; (c) the solutions reach a prescribed terminal entropy profile when time goes to infinity.

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