Pathological solutions of Navier-Stokes equations on $\mathbb{T}^2$ with gradients in Hardy spaces

Antonio Hidalgo-Torn\'e; Jan Burczak

REVIEW 2 major objections 4 minor 25 references

Pathological solutions of Navier-Stokes equations on $\mathbb{T}^2$ with gradients in Hardy spaces

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper proves that p=1 is the sharp boundary between uniqueness and non-uniqueness for very weak 2D Navier-Stokes solutions when vorticity is measured in Hardy spaces, and constructs the p<1 pathology for every smooth initial datum.

desk verdict Sharp p=1 threshold for 2D NSE non-uniqueness in C(H^p) vorticity path spaces — structurally sound, but the new periodic Hardy-space estimates are load-bearing and need careful referee scrutiny. read the letter →

arxiv 2509.08168 v1 pith:ALS7KGBY submitted 2025-09-09 math.AP math.FA

Jan Burczak , Antonio Hidalgo-Torné This is my paper · ORCID

classification math.APmath.FA MSC 35Q3042B3035D30

keywords Navier-Stokesnon-uniquenessHardyspacesveryweaksolutionsvorticityconvexintegrationperiodictoruswell-posednessthreshold2Dfluids

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 tries to show that the index p of the Hardy space H^p marks an exact boundary for uniqueness of very weak two-dimensional Navier-Stokes solutions measured through their vorticity. For p ≥ 1, the class C([0,T],H^p) of vorticity paths is a well-posedness space: two solutions with the same initial datum agree. For any p < 1, the same class is an ill-posedness space: every smooth initial datum admits infinitely many distinct nonzero very weak solutions whose gradients lie in C([0,T],H^p). The proof works by an iterative error-reduction scheme on an approximate Navier-Stokes system, with a newly developed Hardy-space theory on the periodic torus carrying the low-regularity estimates. If true, the result gives a sharp threshold where non-uniqueness disappears and it shows that the right measure of the solution's path space, not just initial data regularity, decides uniqueness.

What carries the argument

The engine is a single iteration step for the Navier-Stokes form of the Reynolds system, where a solution with a smooth stress error R0 is replaced by a new solution whose stress error is arbitrarily small in L1 and whose perturbation gradient is small in H^p. The perturbation is built from fast oscillating, temporally concentrated building blocks, and a crucial point is that the principal corrector is written with enough derivatives so that, after applying the periodic Hardy-space atomic estimates, its gradient lands in H^p for any p<1. Around this step the paper develops the needed periodic Hardy-space theory: maximal-function definition, atomic decomposition, the fact that on the torus at

What would settle it

Compute the H^p norm (with p<1) of the gradient of a concentrated, fast-oscillating corrector of the form used in Section 4, either analytically or numerically on the torus, and check the bound (20): a counterexample where the big-atom estimate of Corollary 3.6 or the projection estimate of Proposition A.17 gives a strictly worse power of the frequency would refute the iteration. More directly, one could search for a periodic function supported on a large ball that is an H^p atom under Definition 3.5 but whose H^p norm is unbounded in the ball's size.

Watch

Extended reading notes

Core claim

On the two-dimensional torus, take any smooth divergence-free initial datum. The paper's central claim is a dichotomy for very weak solutions with vorticity in C([0,T],H^p): for p≥1 the Cauchy problem is unique there, while for every 0<p<1 there are infinitely many distinct very weak solutions, all starting from the same datum, with gradients in C([0,T],H^p). It proves an even stronger flexibility statement: any two smooth solutions can be joined by a single nonzero very weak solution that equals the first on [0,1/8] and the second on [7/8,1]. The threshold is not about the smoothness of the initial value alone: the same smooth datum sits in every H^p, so the deciding object is the path spac

Load-bearing premise

The load-bearing premise is that the newly developed periodic Hardy-space estimates are correct as used—in particular, that on the torus an atom supported on a large ball needs no vanishing moments, and that the divergence-free projection is bounded on H^p; if either estimate fails, the p<1 construction collapses.

Editorial extensions

If this is right

The index p=1 becomes a sharp threshold: measurable vorticity in H^1 is enough for uniqueness, while any p<1 admits non-unique solutions.
Every smooth initial datum supports infinitely many pathological solutions, so non-uniqueness cannot be blamed on rough data.
Any two smooth solutions can be interpolated by a single weak solution that follows one early and the other late—a strong form of flexibility.
The constructed solutions have gradient controlled uniformly in time in H^p and lie in W^{σ,1} for every σ<1, improving the temporal control of previous non-unique 2D Navier-Stokes solutions.
Well-posedness statements for these equations should specify the path space for vorticity, not merely the initial-data space.

Reading between the lines

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

If the periodic Hardy-space estimates carry over to higher dimensions, the same error-reduction step might yield non-unique 3D Navier-Stokes or Euler solutions with vorticity in H^p for p<1; the paper does not claim this.
The H^1 threshold suggests a natural open question: does initial vorticity in H^1 produce a mild solution whose vorticity remains in C([0,T],H^1)? The paper notes this is not addressed.
If the big-atom property is robust, similar constructions could work on other compact manifolds, where atoms on large balls also need no cancellation.
A direct check of gradient estimate (20) on simpler oscillatory test functions would allow the Hardy-space bounds to be tested numerically before relying on the full iteration.

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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

Punchline: this paper claims a clean threshold for 2D Navier-Stokes very weak solutions. For vorticity in C([0,T], H^p), p≥1 gives uniqueness; p<1 gives infinitely many solutions sharing any smooth initial datum, with gradients in C([0,T], H^p) for every p<1. If the periodic Hardy-space machinery holds, this is a sharp result that genuinely upgrades Cheskidov-Luo.

What's new: the threshold statement (Theorem 1.2) and the full-gradient uniform-in-time control (Theorem 1.3). The appendix systematically develops H^p on T^n — atomic decomposition, big-atom principle, moment correction, Leray projector boundedness. That toolbox is useful beyond this paper.

What checks out: I re-ran the parameter bookkeeping in Section 6. The exponent choices satisfy the stated inequalities for fixed p and sigma, and the error terms all decay in λ. The support-preservation step works (the τ_n sum gives the claimed [0,1/8] and [7/8,1]). The nonzero-ness argument using the H^p quasinorm metric is sound. The paper is clear about its dependencies — Buckmaster-Vicol, Cheskidov-Luo, Buck-Modena.

Soft spots, in proportion. Proposition 3.9 has an offhand sentence ('in two dimensions H^1 embeds in (H^1)^*') that is wrong on either reading; the lemma itself is a standard Biot-Savart bound, so this is a carelessness flag, not a fatal one. The main iteration imports its core estimates from the companion Buck-Modena preprint [5], so this paper inherits any issues there. The well-posedness half relies on Cheskidov-Luo's Theorem 1.3 without stating its hypotheses — a little thin, but probably fine. The real soft spot is the new periodic Hardy space theory. The most delicate estimate is the Leray projector bound (Proposition A.17): the transference argument needs the error term E_N to vanish uniformly in x and ζ∈(0,R] as N→∞. The written proof compresses this, and the 'without loss of generality' reductions (smoothness, center, high vanishing moments) are not fully justified. I did not find a definite leak of a power of λ, but the step is load-bearing: if it fails, gradient estimate (20) collapses and the p<1 construction goes with it. This is exactly where a referee should push.

Summary: this is a serious paper with a coherent structure and a significant potential result. It deserves peer review. The decision should hang on a careful check of the appendix, especially A.17 and the moment-correction proposition. I would take it to reading group and cite it once the Hardy-space estimates are verified. Send it out.

Referee Report

2 major / 4 minor

Summary. The paper establishes a sharp dichotomy for very weak solutions of the 2D Navier-Stokes equations on T^2, measured through the path space of vorticity. For p ≥ 1, if two very weak solutions have vorticities in C([0,T], H^p) and agree at time zero, then they coincide (Theorem 1.2(A)). For p ∈ (0,1), the paper claims that for any smooth initial datum there are infinitely many very weak solutions with gradients in C([0,T], H^p), and more generally that any two smooth solutions can be connected by a nonzero solution agreeing with the first on [0,1/8] and with the second on [7/8,1] (Theorems 1.2(B) and 1.3). The construction follows the convex integration scheme for the NS-Reynolds system, with a key iteration Proposition 2.2. The new ingredient is a theory of real Hardy spaces on T^n, developed in the appendix, including atomic decompositions and the boundedness of the Leray projector on H^p(T^n).

Significance. If the periodic Hardy-space toolbox is fully justified, the paper gives a clean and striking threshold: p = 1 is exactly the boundary between uniqueness and non-uniqueness when the vorticity path space is C([0,T], H^p). The convex integration part is standard in structure, but the explicit parameter hierarchy in Section 6 is transparent and checkable. The appendix is a valuable contribution on its own, especially the periodic atomic decomposition and the observation that periodic atoms need not have vanishing moments when supported on large balls. The paper also makes explicit that the ill-posedness phenomenon is a property of the path space, not merely of the initial-data space. No machine-checked proofs are supplied, but the main computation is written in sufficient detail to be audited.

major comments (2)

[Appendix A.5, Proposition A.17] Proposition A.17 is load-bearing: it is used in §4.3.2 to control ∇w_t and hence enters the gradient estimate (20). The transference proof has two gaps. First, the displayed chain after (35) bounds ∥sup_{0<ζ≤R}|Ψζ*Pa|∥_{L^p(T^n)} by N^{-n}∫_{[-N/2,N/2]^n} sup |Ξ_{1/N}(Ψζ*Pa)_ext|^p dx. Since |Ξ_{1/N}|≤1, this integrand is pointwise smaller than |Ψζ*Pa|; to obtain an upper bound one must use periodicity to show that the average of |Ξ_{1/N}|^p over integer shifts is bounded below by a positive constant independent of N. No such argument is supplied. Second, E_N(x)→0 is asserted uniformly for ζ∈(0,R], but the final estimate requires uniformity as R→∞; the large-ζ regime is not treated. The proof of Proposition 3.8 is therefore incomplete, and the p<1 construction depends on it.
[Appendix A.4, Proposition 3.9] Proposition 3.9 is the mechanism used in Theorem 1.2(A) to pass from C_tH^1 vorticity to u∈C_tL^2. The proof invokes the statement 'In two dimensions H^1 ,→(H^1)^*' to control the distribution pairing. As written this is not a standard fact: H^1 denotes both the Hardy space and the Sobolev space, and the asserted embedding is not justified. The desired bound can be obtained directly from H^1⊂L^1, div f=0, the elliptic estimate giving ∇f∈L^1, and the Sobolev embedding W^{1,1}(T^2)↪L^2. The conclusion is correct, but the present proof needs to be replaced or substantially clarified.

minor comments (4)

[§4.3.2, final estimate] The text says 'Remark 3.6 and (19)' but no Remark 3.6 exists; the relevant result is the embedding L^s⊂H^p for s>1 (Proposition 3.3 together with Lemma A.4). Please cite the correct statement.
[Theorem 1.2(B) proof] The proof that there are 'infinitely many' distinct solutions is only implicit. Since Theorem 1.3 gives one solution for each choice of v2, one should explicitly argue that different choices of v2 (e.g., different final data) produce different solutions on [7/8,1] and hence distinct u. Please make this step explicit.
[Appendix A.4, notation] In the proof of Proposition 3.9, the same symbol H^1 is used for the Hardy space and for the Sobolev space H^1(T^2). This is confusing and should be resolved by notation or a sentence.
[Throughout] Minor typos: 'a ill-posedness space' should be 'an ill-posedness space'; §4.3.2 refers to 'Corollary 3.8' when Proposition 3.8 is meant; the proof of Proposition A.17 overloads P for both periodic and full-space Leray projectors.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the p=1 threshold is assembled from an external uniqueness theorem and an external convex-integration construction; the periodic Hardy-space appendix is new, not self-referential.

full rationale

The paper's derivation chain does not reduce any of its claims to its own assumptions. The well-posedness side (Theorem 1.2(A)) depends on the external Ladyzhenskaya-Prodi-Serrin uniqueness criterion as stated in Cheskidov-Luo [12], after a short reduction using Proposition 3.9. Proposition 3.9 is proved in Appendix A.4 from a Helmholtz-style decomposition; regardless of whether that proof is fully detailed, it does not presuppose the uniqueness being proved. The ill-posedness side (Theorem 1.2(B), Theorem 1.3) is a convex-integration iteration built on Proposition 2.2. The gradient-in-H^p estimate (20) is derived by applying the periodic Hardy-space tools of the appendix (Corollary 3.6, Proposition 3.7, Proposition 3.8/A.17) to explicit correctors. Those appendix results are proved from the maximal-function and atomic definitions in the paper, not assumed from the target theorem. There is no fitted parameter renamed as a prediction: the iteration parameters are chosen explicitly and all error terms are shown to be small by negative powers of λ. The paper does cite the first author's earlier work [10] for the antidivergence Lemma 3.1, but that is a published, independent technical lemma and not the source of the threshold dichotomy. The Skeptic's concern about the correctness of Corollary 3.6, Proposition 3.7, and Proposition A.17 is a verification risk, not circularity: the estimates are not equivalent to the theorem's conclusion by construction, and no displayed equation in the appendix assumes the p<1 non-uniqueness result. The paper's own footnote 2 noting that existence of mild solutions for p=1 is open is a limitation, not a circular step. Therefore no circularity step can be exhibited, and the circularity score is 0.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The central dichotomy rests on two pillars: the convex integration machine, which imports the building-block, corrector-splitting, and decorrelation estimates from Buck-Modena [5] and the framework of Buckmaster-Vicol and Cheskidov-Luo, and the new periodic Hardy space theory developed in the appendix from classical Stein/Folland-Stein material. The only new objects are mathematical definitions (periodic atoms, representative balls), not empirically fitted quantities. The Section 6 parameter hierarchy is a set of proof-construction knobs with explicit feasibility conditions for every p and sigma in the theorem statements.

free parameters (1)

Convex integration parameter hierarchy (exponents alpha, a, b, beta, gamma and integrability s) = a=5, b=11, beta=4, gamma=1, s close to 1, alpha large (alpha > 2/p, plus Section 6 inequalities)
Section 6 fixes mu1 = lambda^alpha, mu2 = lambda^{alpha+5}, kappa^{1/2} = lambda^4, omega = lambda^{alpha+11}, nu = lambda and requires alpha large enough that all displayed error exponents are negative for the fixed p in (0,1) and sigma in (0,1). Feasible for every p and sigma; the bookkeeping checks out against Sections 4-5. These are construction parameters, not empirical fits, listed here for

assumptions (7)

domain assumption LPS-type uniqueness theorem for very weak 2D NSE solutions (Theorem 1.3 of Cheskidov-Luo [12])
Invoked in Section 1 to prove Theorem 1.2(A): u in C([0,T], L^2) from Proposition 3.9 puts both solutions in the LPS class, yielding uniqueness. The precise hypotheses of [12, Theorem 1.3] are not quoted, so a referee should confirm they cover exactly this class.
domain assumption Global smooth solvability of 2D NSE for smooth periodic data
Footnote 2 and the proof of Theorem 1.3 need the smooth solutions v1, v2 in the interpolation statement; standard 2D NSE well-posedness for smooth data.
domain assumption Building-block estimates (7), corrector splitting (Proposition 6.1), and decorrelation (Lemma 7.2, 7.3) from Buck-Modena [5]
Sections 4.1.1, 4.2, 4.3.1 import the core estimates of the iteration from the companion preprint [5] rather than proving them; if any of those estimates miss a factor, the exponent balance in Section 6 has to be redone.
standard math Classical H^p(R^n) theory of Stein [23] and Folland-Stein [19] (maximal function equivalence, atomic decomposition)
Appendix A adapts these to the torus; Propositions A.3, A.6, A.10, A.15 rest directly on the classical arguments.
domain assumption Antidivergence operators of Burczak-Modena-Szekelyhidi [10, Proposition 4] (Lemma 3.1)
Section 3.1; used to invert divergence in the error estimates of Section 5. Duplicated authorship with the present paper, but the result is published and external to this construction.
standard math Nash-type decomposition of near-identity symmetric matrices into rank-one pieces (Lemma 3.2)
Section 3.1, from Brie-Colombo [3, Section 5]; standard convex integration tool.
standard math Hodge decomposition with W^{1,2} potentials and the 2D embedding W^{1,2} into BMO
Appendix A.4, proof of Proposition 3.9. The text writes 'H^1 embeds in (H^1)^*', which is at best a compressed way of saying curl f in H^1 (Hardy) pairs against g1 in W^{1,2} subset BMO; stated cleanly this is standard.

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Pith. "Pith review of Pathological solutions of Navier-Stokes equations on $\mathbb{T}^2$ with gradients in Hardy spaces." pith.science (2026). https://pith.science/paper/ALS7KGBY

@misc{pith2026250908168,
  author       = {Pith},
  title        = {Pith review of: Pathological solutions of Navier-Stokes equations on $\mathbbT^2$ with gradients in Hardy spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ALS7KGBY}},
  note         = {Machine review of arXiv:2509.08168}
}

abstract

For an arbitrary smooth initial datum, we construct multiple nonzero solutions to the $2$d Navier-Stokes equations, with their gradients in the Hardy space $\mathcal{H}^p$ with any $p \in (0,1)$. Thus, in terms of the path space $C(\mathcal{H}^p)$ for vorticity, $p=1$ is the threshold value distinguishing between non-uniqueness and uniqueness regimes. In order to obtain our result, we develop the needed theory of Hardy spaces on periodic domains.

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Pith tools

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

[1] [5]

Compactly supported anomalous weak solutions for 2D Euler equations with vorticity in Hardy spaces

M. Buck and S. Modena. Compactly supported anomalous weak solutions for 2D Euler equations with vorticity in Hardy spaces.ArXiv 2405.19214, 2024

work page Pith review arXiv 2024

[2] [1]

Albritton, E

D. Albritton, E. Bru´ e, and M. Colombo. Non-uniqueness of Leray solutions of the forced Navier-Stokes equations.Ann. of Math. (2), 196(1):415–455, 2022

2022

[3] [2]

Albritton, E

D. Albritton, E. Bru´ e, M. Colombo, C. De Lellis, V. Giri, M. Janisch, and H. Kwon. Instability and non-uniqueness for the 2D Euler equations, after M. Vishik, volume 219 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2024

work page 2024

[4] [3]

Bru´ e and M

E. Bru´ e and M. Colombo. Nonuniqueness of solutions to the Euler equations with vorticity in a Lorentz space.Communications in Mathematical Physics, 403(2):1171–1192, 2023

work page 2023

[5] [4]

Bru` e, M

E. Bru` e, M. Colombo, and A. Kumar. Flexibility of two-dimensional euler flows with integrable vorticity.Preprint ArXiv:2408.07934, 2024. 29

arXiv 2024

[6] [6]

Buck and S

M. Buck and S. Modena. Non-uniqueness and energy dissipation for 2D Euler equations with vorticity in Hardy spaces.Journal of Mathematical Fluid Mechanics, 26(2):1–39, 2024

work page 2024

[7] [7]

Buckmaster, M

T. Buckmaster, M. Colombo, and V. Vicol. Wild solutions of the Navier-Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1.J. Eur. Math. Soc. (JEMS), 24(9):3333–3378, 2022

2022

[8] [8]

Buckmaster and V

T. Buckmaster and V. Vicol. Nonuniqueness of weak solutions to the Navier-Stokes equa- tion.Ann. of Math. (2), 189(1):101–144, 2019

work page 2019