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REVIEW 4 major objections 4 minor 39 references

An Onsager type theorem for the Euler-Boussinesq equations in two spatial dimensions

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Pith's one-line read The 2D Euler-Boussinesq equations admit Hölder-continuous weak solutions for every exponent below 1/3, matching the Onsager threshold for Euler flows.

desk verdict Sharp 1/3 threshold for 2D Euler-Boussinesq is the right target and the setup is credible, but the closing inequality in Corollary 6.24 does not close as written. read the letter →

arxiv 2502.04803 v1 pith:GIV6BLAR submitted 2025-02-07 math.AP math-phmath.MP

Ujjwal Koley This is my paper · ORCID
classification math.APmath-phmath.MP MSC 35Q3135D3076B03
keywords Euler-BoussinesqequationsOnsagerconjectureconvexintegrationNashiterationHölderweaksolutionstemperatureLpnonconservation2Dincompressibleflowscompacttemporalsupport
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

This paper claims that the two-dimensional inviscid Euler-Boussinesq system—the model of convection coupling a velocity field and a temperature field—is as flexible as the Onsager conjecture predicts for Euler: for every Hölder exponent γ<1/3 there exist nontrivial weak solutions with compact temporal support, in C^γ(R×$T^{2}$) for both velocity and temperature. Such solutions violate conservation of the temperature's L^p-norm and hence the classical identities (1.2). If true, the result closes the flexibility gap left by earlier 2D constructions, whose Hölder exponents fell far short of 1/3, and matches the Onsager-critical threshold previously established for Boussinesq in three dimensions. The proof is an iterative Newton–Nash scheme in which a Newton step solves a linearized Boussinesq system with temporally oscillatory forcing and a Nash step adds highly oscillatory shear perturbations, with the velocity and temperature error cancellations handled jointly.

What carries the argument

The key machinery is a two-family version of the Newton–Nash iteration developed for the 2D Euler equations: Lemma 4.1 supplies one family of 1-periodic functions g_{ξ,p,n} for velocity amplitudes and a second family h_{ζ,p,n} for temperature amplitudes whose temporal supports are pairwise disjoint across both families, so the velocity and temperature building blocks never interact during error cancellation. Around this, the Newton step solves a linearized Euler-Boussinesq system with temporally oscillatory forcing, using a pair of stream functions for the velocity and temperature perturbations, while the Nash step decomposes the velocity perturbation into one component cancelling the velocity stress R_q and another cancelling the velocity–temperature transport error T_q, using the two geometric lemmas Lemma 2.4 and Lemma 2.5. Repeating over Γ=⌈1/(1/3−β)⌉ Newton steps and then one Nash step at each iteration level drives the errors to zero while preserving the target Hölder regularity, with a small exponent loss absorbed by the choice b<(1+3β)/6β.

What would settle it

Inspect the proof of Lemma 4.1 and look for indices (ξ,p,n) and (ζ,q,m) with supp g_{ξ,p,n}∩supp h_{ζ,q,m}≠∅; if such an overlap exists, the products $g^{2}$ $h^{2}$ appearing in the oscillation error do not vanish identically, and the error bound in Lemma 6.12 loses its gain of λ_{q+1}^{−(1−2α)}, which would exceed the allowed Reynolds stress tolerance δ_{q+2}λ_{q+1}^{−2α}.

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Extended reading notes

Core claim

The central discovery is that the basic obstruction to Onsager-critical regularity in two dimensions—two non-parallel plane-wave directions must intersect—can be circumvented for the coupled velocity–temperature system by using oscillations in time. The paper constructs, for any 0≤γ<1/3, a sequence of smooth solutions of the Euler-Boussinesq-Reynolds system whose Reynolds stresses R_q and T_q tend to zero, with the increments v_{q+1}−v_q and θ_{q+1}−θ_q small in C^γ; the limit is a weak solution of (1.1) with compact temporal support that fails to conserve the temperature's L^p-norm. The main theorem, Theorem 1.2, states exactly this existence in C^γ(R×$T^{2}$)×C^γ(R×$T^{2}$) for every γ<1/3.

Load-bearing premise

The whole construction rests on being able to schedule the fast time oscillations of the velocity pieces and the temperature pieces so that they never occur at the same moment; the paper borrows this scheduling from a lemma proved for one family, and if the two schedules overlap the error cancellation breaks.

Editorial extensions

If this is right

  • The 2D Euler-Boussinesq system has the same Onsager-critical flexibility threshold as the 3D case: weak solutions exist at every Hölder exponent below 1/3 and fail to conserve temperature norms.
  • Because the paper observes that any 2D solution extends trivially to a k-dimensional solution for k≥2, the construction transfers Onsager-critical nonconservative solutions to all higher spatial dimensions.
  • The temperature L^p identity (1.2) is violated for all p≥1 by compactly supported-in-time solutions, so the dimension-independent rigidity part—conservation above 1/3—is sharp in two dimensions.
  • The Newton–Nash splitting of errors into velocity-stress and velocity–temperature transport components provides a template for other coupled active-scalar systems in 2D where the scalar and velocity building blocks must be kept disjoint.

Reading between the lines

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

  • The paper leaves open what happens at exactly γ=1/3, so a natural next step suggested by the argument is to determine whether flexibility persists at the critical exponent or conservation is restored, as in the Euler case.
  • If the mutual disjointness of the g- and h-families in Lemma 4.1 is written out in full, the same two-family temporal partition could plausibly be reused for other coupled systems whose linearization is a well-posed transport-type problem.
  • A concrete testable extension would be a numerical simulation of the first few Newton–Nash levels to see whether gains of λ_{q+1}^{−(1−2α)} genuinely require exact support disjointness or whether sufficiently rapid temporal decay of the building blocks would suffice.
  • One could attempt to replace the deferred single-family support lemma with a self-contained construction in the two-family setting, which would remove the main external ingredient on which the Newton and Nash cancellations currently rest.
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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 / 4 minor

Summary. The paper constructs non-trivial weak solutions (v,θ) of the 2D incompressible Euler-Boussinesq system with compact temporal support and Hölder regularity C^γ × C^γ for every γ<1/3, thereby reaching the same threshold as Onsager's conjecture for Euler. The proof is a Newton–Nash iterative scheme: Proposition 3.1 is the inductive statement; Section 5 performs Γ Newton steps based on a linearized Boussinesq system with oscillatory forcing; Section 6 performs the Nash step with geometric lemmas, stationary-phase estimates, and a material mollification; Corollary 6.24 closes the estimates; the main theorem follows by a spatial-temporal mollification argument. The argument is structurally coherent, with explicit parameter constraints and a self-contained linear well-posedness appendix, but several lemmas that are new to the Boussinesq system are deferred to [21] or omitted.

Significance. If the deferred estimates are completed, the result is significant: it would be the first Onsager-critical (1/3−) flexibility theorem for the 2D Boussinesq system, improving the exponents of Tao–Zhang and matching the 3D flexibility result of Miao–Nie–Ye, and it would demonstrate that the 2D time-oscillation Newton–Nash method of Giri–Radu extends to active-scalar systems. The manuscript is transparent about its dependencies: it states the parameter ranges explicitly, gives the base case for the induction, and provides a full well-posedness proof for the linearized system in Appendix A. There are no machine-checked proofs or reproducible code, but the structure is checkable in principle; the main obstruction is the number of deferred lemmas that are genuinely new in the Boussinesq setting.

major comments (4)
  1. [Lemma 4.1 (Section 4)] Lemma 4.1 asserts the existence of two families g_{ξ,p,n} and h_{ζ,p,n} with pairwise disjoint temporal supports, including across the two families, and the proof is deferred to [21, Lemma 3.3] 'modulo cosmetic changes'. The cited lemma treats only a single family of building blocks; the mutual disjointness of the g- and h-families is a new requirement that is used in Section 5 (e.g., in the definitions of S_{q,Γ} and X_{q,Γ} and in the disjointness of the sums in (6.10)–(6.11)) to prevent velocity–temperature interactions. Since no construction is supplied, the Newton step is not fully proved as written. The gap is likely repairable (a finite partition of the period into 2Γ(|Λ_R|+|Λ_T|) disjoint intervals with unit-L^2-scaled bumps would suffice), but it should be written out in the manuscript.
  2. [Lemma 6.10 (Section 6.5)] Lemma 6.10, which bounds the buoyancy error R(θ^{(s)}_{q+1}e_2) and its material derivative, is a genuinely new term for the Boussinesq system and its proof is omitted ('For the sake of brevity, we skip the details'). These bounds enter directly into the closing estimate Corollary 6.24: the displayed sum for R_{q+1}, T_{q+1} includes the δ^{1/2}_{q+1}λ^{-(1-α)}_{q+1} term, and the material-derivative sum includes the ℓ^{-1}_{t,q}δ^{1/2}_{q+1}λ^{-(1-5α)}_{q+1} term. If either estimate loses one power of the frequency or of ℓ^{-1}_{t,q}, the induction fails. Because this term is not present in [21], a citation to the Euler proof is not sufficient, and a full derivation is required.
  3. [Section 6, Lemmas 6.8 and 6.15–6.18] Several additional lemmas in Section 6 are asserted with proofs deferred to [21]: Lemma 6.8 (Nash error with ∇θ̄_{q,Γ}), Lemmas 6.16–6.17 (temperature oscillation errors), and Lemma 6.18 (divergence–mean corrector error for the temperature). These terms are new in the Boussinesq setting, and the cancellation mechanism for T_{q+1,O} via the h-family building blocks (Section 6.7) has no analogue in the Euler equations. Since Corollary 6.24 is obtained by summing exactly these contributions, the manuscript should state which estimates from [21] and [24] apply verbatim and which require modification; the blanket statement 'modulo cosmetic changes' is not verifiable as written.
  4. [Corollary 6.24 proof] The closing argument for the material derivative estimates relies on the chain '1 < b < 1+3β/(6β) < 1/(3β)'. This chain is false for β>2/9, since 1+3β/(6β)=3/2 while 1/(3β)<3/2 in that range, and Proposition 3.1 allows all β∈(0,1/3). A direct exponent comparison is needed to justify the displayed absorption into δ_{q+2}δ^{1/2}_{q+1}λ^{N+1−2α}_{q+1}; the present text does not provide it for the full stated parameter range.
minor comments (4)
  1. [Section 3.2, base case] The verification of the base case states 'by our choice (2b−1)β<1/3', but this is not implied by the assumptions b<1+3β/(6β) and β<1/3; the condition actually needed for the displayed inequality is (2b−1)β<1/2. The argument still works for sufficiently large a, but the stated reason is inaccurate.
  2. [Lemma 4.1 statement] The statement of Lemma 4.1 uses the same symbol p (and later q) for both the parity index and the iteration level q, which is confusing. For example, 'whenever pξ,p,nq ≠ pζ,q,mq' mixes the parity labels with the iteration index; a separate notation such as ε∈{e,o} would improve readability.
  3. [Section 5, notation] In Section 5, the operator R∇K is used as a Calderón-Zygmund-type operator before it is explicitly defined; please define it at first use, since it is central to the stream-function formulation (5.26)–(5.27).
  4. [Section 5.1, reliance on [24]] The definitions of the Newton perturbation amplitudes a_{ξ,k,n} and b_{ζ,k,n} in (5.2)–(5.3) are attributed to the submitted preprint [24]. Since that paper is not yet published, the present manuscript should restate the relevant ansatz or clearly identify exactly which estimates from [24] are being reused, so that the reader can verify the argument without access to the preprint.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the construction is an explicit Newton-Nash induction; the only mild self-citation ([24]) is motivational and not load-bearing.

full rationale

The paper's central claim (Theorem 1.2) is an existence theorem proved by an inductive Proposition 3.1. The error terms (R_{q+1}, T_{q+1}) in Corollary 6.24 are bounded by explicit powers of the frequency and amplitude parameters; the constraints on beta, b, alpha, a are chosen to make those inequalities close, not fitted to the target conclusion. There is no equation in which the conclusion (existence of C^gamma solutions) is assumed as an input. Deferred results are not circular reductions: Lemma 4.1 is stated as a finite partition construction and its proof is deferred to [21, Lemma 3.3] 'modulo cosmetic changes'; even if the mutual two-family disjointness is not literally in [21], it is a repairable construction issue, not a self-referential definition. The Boussinesq coupling estimates (Lemmas 6.8-6.10, 6.15-6.18) are new estimates for terms specific to the temperature equation; if one of them misses a power of lambda, Corollary 6.24 fails, but that is a correctness risk, not circularity. The self-citation [24] (Giri-Koley, submitted) is invoked for the design of the perturbation ansatz ('similar strategies as in [24]'), but the present paper writes the explicit formulas (5.2)-(5.3), (6.10)-(6.13) and proves the needed estimates. The central existence argument therefore does not reduce to a conclusion of [24] or of the author's prior work. Consequently no circular step is present; at most there is a minor, non-load-bearing self-citation and several unverified estimates that should be checked by referees.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on standard convex integration background, which the paper cites, and on one new construction-specific assumption, Lemma 4.1, whose proof is deferred. There are no empirical free parameters; all constants in the proof are chosen from open ranges and the existence claim is robust to those choices.

assumptions (3)
  • standard math Hölder interpolation, mollification and commutator estimates (Propositions 2.1-2.3 and 2.8), stationary phase (Proposition 2.7), Calderón-Zygmund boundedness (Propositions 2.10-2.11), and transport estimates (Proposition 2.9).
    Taken as background from the cited literature; used throughout Sections 4 to 6.
  • standard math Geometric decompositions: Lemma 2.4 (symmetric matrices as sum of squares of rank-one matrices) and Lemma 2.5 (any vector in R^2 decomposed into three directions with affine coefficients), with proofs referenced to [38,20].
    These lemmas are the building blocks for inverting the Reynolds stress and transport error; they are standard in convex integration but imported here.
  • ad hoc to paper Lemma 4.1: existence of two families of smooth 1-periodic functions g_{ξ,p,n} and h_{ζ,p,n} with unit L^2 norm and pairwise disjoint temporal supports, including across families.
    This is the key non-interaction property for the 2D construction. The proof is deferred to [21, Lemma 3.3] 'modulo cosmetic changes', but the mutual disjointness of the g- and h-families is new and not explicitly proven.

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Pith. "Pith review of An Onsager type theorem for the Euler-Boussinesq equations in two spatial dimensions." pith.science (2026). https://pith.science/paper/GIV6BLAR

@misc{pith2026250204803,
  author       = {Pith},
  title        = {Pith review of: An Onsager type theorem for the Euler-Boussinesq equations in two spatial dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GIV6BLAR}},
  note         = {Machine review of arXiv:2502.04803}
}
abstract

In this article, we construct non-trivial weak solutions $(v, \theta)$ to the inviscid Euler-Boussinesq system in two spatial dimensions. These solutions exhibit compact temporal support, thereby violating the conservation of the temperature's $L^p$-norm. Furthermore, the pair $(v, \theta)$ resides in the H\"older space $C^\gamma(\mathbb R \times \mathbb T^2) \times C^\gamma (\mathbb R \times \mathbb T^2)$ for any exponent $\gamma<1/3$. The methodology integrates a Nash iteration scheme with a linear decoupling technique to achieve these results.

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