On non-uniqueness of mild solutions and stationary singular solutions to the Navier-Stokes equations

Alexey Cheskidov; Hedong Hou

REVIEW 3 major objections 4 minor 1 cited by

On non-uniqueness of mild solutions and stationary singular solutions to the Navier-Stokes equations

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Unconditional uniqueness of Navier-Stokes mild solutions fails in every Besov space with negative regularity index, even subcritical ones, by constructing non-trivial stationary singular solutions via convex integration.

desk verdict Main theorem (non-uniqueness in every negative-regularity Besov space) is a real advance and mostly well-proved; the L^2/small-α results have an unproved non-zeroness step, and the 'mild solution' convention is worth remembering. read the letter →

arxiv 2603.03666 v2 pith:YYEA3UG3 submitted 2026-03-04 math.AP

Alexey Cheskidov , Hedong Hou This is my paper

classification math.AP MSC 35A0235K5535Q3042B3776D05

keywords Navier-StokesequationsunconditionaluniquenessmildsolutionsBesovspacesconvexintegrationstationarysingularfractionalparaproduct

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 proves that for the Navier-Stokes equations on the torus, unconditional uniqueness of mild solutions fails in all Besov spaces with negative regularity index, including subcritical spaces where local well-posedness holds. For any θ>0 and q,r∈[1,∞], one can find arbitrarily small initial data in B^{-θ}_{q,r} that admit two distinct mild solutions in C([0,T];B^{-θ}_{q,r}). The construction relies on building non-trivial stationary singular solutions—time-independent solutions that solve the stationary equation with the nonlinear term defined as a paraproduct—and then converting them into mild solutions. The paper extends the result to fractional Navier-Stokes equations with arbitrarily large powers of the Laplacian, and proves the opposite side: uniqueness of stationary weak solutions in an endpoint critical space.

What carries the argument

The central tool is convex integration with Mikado flows, adapted to the stationary Navier-Stokes-Reynolds system div(u⊗u)+(-Δ)^α u+∇p=div R. Each iteration adds a high-frequency, highly concentrated velocity perturbation w whose Fourier support lies in a thin annular shell far above the previous frequencies. The nonlinearity u⊗u is defined not as a classical pointwise tensor product but as a paraproduct in H^{-s} for distributions modulo constants, because the constructed solutions need not lie in L^2_loc. A key arithmetic lemma ensures that the oscillatory factors σ_kk^⊥ all land in the same dyadic shell, so the perturbation stays divergence-free and its Fourier support is controlled. A fi

What would settle it

Take one of the constructed stationary singular solutions u and compute the classical (distributional) tensor product u⊗u in D'(T^d). If u⊗u actually belongs to L^1_loc, then the paraproduct definition would coincide with the classical product and the non-uniqueness would survive the stricter classical notion. If, on the other hand, u⊗u is not locally integrable, then the claim 'two mild solutions in C([0,T];B^{-θ}_{q,r})' holds only under the paper's extended definition; verifying which of these occurs for an explicit seed (e.g., a simple trigonometric polynomial) would settle whether the res

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for every dimension d≥2, every θ>0, and every q,r∈[1,∞], there exists divergence-free initial data u_in ∈ B^{-θ}_{q,r} with arbitrarily small norm such that two mild solutions u and v, both in C([0,T];B^{-θ}_{q,r}), satisfy u(0)=v(0)=u_in but u(t)≠v(t) for every t∈(0,T]. The proof constructs non-trivial 'singular solutions' to the stationary fractional Navier-Stokes equations via convex integration. These are distributions u with zero mean and zero divergence that satisfy the stationary equation with u⊗u defined as a paraproduct in a negative Sobolev space H^{-s}; such a u automatically gives a time-independent mild solution. Starting from a nonzero smooth s

Load-bearing premise

The entire construction relies on defining the nonlinear term u⊗u as a paraproduct in a negative Sobolev space H^{-s} for distributions that need not be locally integrable; if 'mild solution' is required to mean the classical integral equation with u⊗u ∈ L^1_loc, the constructed stationary singular solutions are not valid mild solutions and the non-uniqueness statement does not follow.

Editorial extensions

If this is right

If the central claim is correct, the Navier-Stokes Cauchy problem is locally well-posed but not unconditionally well-posed in every subcritical Besov space with negative regularity index.
The failure of unconditional uniqueness extends to fractional Navier-Stokes equations with arbitrarily large α, in both Besov and (for α>(d+2)/4) Lebesgue spaces L^p with p<2.
For 0<α<(d+1)/4, the construction yields non-uniqueness of weak solutions in L^2 with arbitrarily small initial data, giving infinitely many distinct weak solutions.
The uniqueness half of the paper shows that stationary weak solutions in the endpoint critical space B^{-1}_{∞,1} (or its fractional analogue) must be trivial, which contrasts with the abundance of singular stationary solutions in all negative Besov spaces.
The result sharpens the known ill-posedness boundary: below B^{-1}_{∞,∞} there are no unconditional uniqueness classes at all.

Reading between the lines

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

The paper's non-uniqueness is tied to an extended notion of mild solution where u⊗u is defined as a paraproduct; if one insisted on classical mild solutions with u⊗u ∈ L^1_loc, the stationary singular solutions would not qualify, so the result would not contradict the classical Fabes-Jones-Rivière uniqueness in L^p.
The stationary singular solutions are time-independent, so they represent eternal 'background' flows that coexist with smooth solutions; this suggests that in negative-regularity data, the mere specification of the initial datum does not determine the evolution unless one also fixes the product law for distributions.
One can test the construction's robustness by asking whether the same iteration works for the classical Navier-Stokes equations in L^2-based spaces; the Fourier-shell separation and paraproduct estimates suggest that any such attempt would need a genuinely L^2-integrable product, which the current non-L^2 solutions fail.
The endpoint uniqueness result (Theorem 1.7) implies that the stationary non-uniqueness phenomenon is confined to spaces more singular than the critical line; identifying the exact critical/marginal regularity where non-trivial stationary solutions first appear is a natural next step.

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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

Main takeaway: the central result—failure of unconditional uniqueness in all negative-regularity Besov spaces, including subcritical ones—is new and, as far as I can tell, correct. The proof is a substantial convex-integration construction, and the authors earn their claim. I'd send this to a serious referee on the strength of Theorem 1.1 alone.

The genuinely new part is the extension of non-uniqueness to every Besov space B^{-θ}_{q,r}, θ>0, and to fractional Navier-Stokes with arbitrarily large dissipation. The construction of stationary singular solutions via a frequency-localized iteration is clean; Proposition 4.1's Fourier-support separation is what makes the nonzero limit work, and the estimates in Sections 7–8 are carefully done. The endpoint uniqueness result (Theorem 1.7) is a nice complement.

Now the soft spots. The L^2 results—Theorem 1.5(ii) and Corollary 1.6, relying on Proposition 4.2—have a hole. In the Besov case, the Fourier localization gives you \hat u(m)=\hat u_0(m) for |m|=1, which forces the limit to be nonzero. In the L^2 iteration, Proposition 4.2 has no frequency-localization conclusion, and the proof just says "one can mimic the above proof." Mimicking the Besov argument doesn't work without an analogue of the low-mode preservation. The constructed sequence could in principle converge to zero, and then the "non-trivial" stationary solution would be trivial. I did not find any argument in the text ruling this out. This is not a fatal flaw for the paper's main message, but the L^2 claims are not fully supported as written.

The other thing to keep in mind is the solution concept. The authors define mild solutions using a paraproduct tensor product, valid for distributions even when u⊗u is not locally integrable. They flag this in a footnote and call such solutions "singular mild solutions." That's an honest choice, but it means Theorem 1.1 is a statement about this extended class, not about classical mild solutions with u⊗u ∈ L^1_loc. If the intended audience cares about the classical notion, the title overreaches slightly.

Overall: recommended for peer review. The main theorem deserves a serious referee; the L^2 section needs a repaired non-zeroness argument and possibly a small rewrite. I'd want to see that fixed before accepting the whole package.

Referee Report

3 major / 4 minor

Summary. The paper proves failure of unconditional uniqueness of mild solutions to the (fractional) Navier-Stokes equations in every negative-regularity Besov space B^{-θ}_{q,r} on the torus, for any θ>0 and d≥2. The strategy is to construct non-trivial stationary singular solutions via convex integration with intermittent Mikado flows and Fourier localization. The central Besov construction (Propositions 4.1, 4.2 and Sections 6–8) is written in considerable detail: a sequence of approximate solutions (u_n, R_n) to the stationary Navier-Stokes-Reynolds system is produced, with the velocity perturbation supported in a high-frequency annulus separated from the previous frequencies, giving both convergence in all negative Besov spaces and a nontrivial limit. The paper also states L^2-based existence results for small α (Theorem 1.5(ii), Corollary 1.6) and a uniqueness theorem for stationary weak solutions in an endpoint critical space (Theorem 1.7). The Besov iteration is largely coherent, but several load-bearing steps in the L^2 claims and in the proof that the constructed data are non-smooth are missing or rely on inapplicable results.

Significance. If the main theorem (Theorem 1.1) is fully correct, it is a major advance: unconditional uniqueness of mild solutions had been known in subcritical Lebesgue spaces and in critical/supercritical Besov spaces only for classical solution classes, and this would be the first non-uniqueness result in subcritical negative-regularity Besov spaces. The method is original in combining convex integration with frequency localization and intermittency, and the Besov part of the paper is detailed and reproducible, with explicit estimates. The uniqueness theorem (Theorem 1.7) is also a worthwhile contribution. However, the paper's scope is limited by its definition of mild solutions via a paraproduct in H^{-s} rather than a classical L^1 tensor product, and the L^2/second-solution results as written are not fully justified.

major comments (3)

[§5.1.2, Proposition 4.2] The proof of Theorem 1.5(ii) does not establish that the limiting L^2 solution u is nonzero. Proposition 4.2 contains no Fourier-support separation: the perturbation in Section 9, w = Σ a_k W_k(γx) + ..., has frequencies that are not confined to a high annulus, so the low modes of u0 can be altered. The sentence 'One can mimic the above proof' (end of §5.1.2) is insufficient, because the Besov proof in §5.1.1 used the Fourier separation to show \hat u(m)=\hat u_0(m) for |m|=1, and the L^1 bound ∥u-u0∥≤1; neither is available here. Without a lower bound on some Fourier mode of the limit (or an alternative argument), the sequence could converge to zero, in which case the constructed 'non-trivial' stationary solution is trivial and Corollary 1.6 collapses.
[§5.2, proof of Theorem 1.2(i)] The argument 'u_in /∈ C∞. Otherwise, Corollary 3.1 implies that u_in = 0' is not valid for the data constructed in that proof. Corollary 3.1 requires u∈L^p with p in the subcritical range (e.g. p>d/(2α-1) for α≤(d+2)/4). However, the iteration in Theorem 1.2(i) uses p_n=3/2, so the limit u_in is only guaranteed to lie in L^{3/2}. For α=1 and d≥2, 3/2 is strictly below the critical exponent d/(2α-1)=d, so Corollary 3.1 does not apply. The conclusion that u_in is non-smooth needs a different proof; for example, the direct energy identity for a smooth stationary solution, ∥(-∆)^{α/2}u∥_{L^2}^2 = ∫(u⊗u):∇u = 0, shows any smooth mean-zero stationary solution is zero. This is a local fix, but it must be made explicitly.
[Corollary 1.6] Corollary 1.6 is stated without proof and does not follow from Theorem 1.5(ii) alone. For 0<α<(d+1)/4, the L^2-critical condition α>(d+2)/4 fails, so the local well-posedness result of Proposition 2.1 is not available in L^2 (or in any L^p with p≤2, since d/(2α-1)>2). The stationary solution U(t)=u_in provides only one weak solution; the existence of a second weak solution with the same L^2 initial data is not established anywhere in the paper. Either a second solution must be constructed (e.g., via the same convex integration) or the corollary must be removed or qualified.

minor comments (4)

[§5.2, embedding reduction] The reduction to a smaller θ near the start of §5.2 is written in a confusing manner. The embedding B^{-θ}_{q,1} ↪ B^{-θ_1}_{q_1,r_1} holds for θ_1≥θ, so to cover all θ>0 one must construct solutions for θ arbitrarily small; the text should state this clearly.
[§5.2, choice of u0] The inequality ∥u0∥_{B^{-θ}_{∞,1}} ≤ ∥u0∥_{L∞} in the proof of Theorem 1.2(i) is missing a constant depending on θ (since ∑_{N≥1} N^{-θ} is finite but not equal to 1). This is harmless—one can rescale u0—but the estimate as written is not exact.
[§1.3, Definition 1.3] The paper correctly emphasizes that u⊗u is defined via paraproduct in H^{-s} and that the solutions may not be classical mild solutions with locally integrable nonlinearity. This point should also be reflected in the abstract and introduction, as it is essential for interpreting Theorem 1.1.
[§1.3, line after Theorem 1.5] There is a small formatting/typo issue: 'extended it to Tϵ∈(0,1) L2−ϵ∩ ˙H−ϵ(T2)' is unclear; the intended expression should be typeset properly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the non-uniqueness construction is a self-contained convex-integration argument; the main caveat is the extended paraproduct definition of u⊗u, which is a modeling choice rather than a circular reduction.

full rationale

The paper's central derivation chain is: (i) construct nontrivial stationary singular solutions to NSα by an iterative convex-integration scheme (Propositions 4.1 and 4.2); (ii) show via Proposition 1.4 that any such solution yields a constant mild solution; (iii) combine with the independently proved local well-posedness theory (Proposition 2.1) to obtain a second, smooth mild solution from the same initial data. None of these steps assumes the target non-uniqueness. The extended definition of u⊗u as a paraproduct in Ḣ^{-s} (Definition 1.3, footnote p.2) is an explicit modeling choice: the paper proves the stationary equation in that sense and does not redefine the conclusion into existence. Cited results such as [CL22], [CCFS08], [CD14], and [LR25] are prior published theorems or techniques with independent proofs; even though some share an author with the present paper, they are not used as unverified assertions of the present conclusion. The L² iteration for Theorem 1.5(ii)/Corollary 1.6 lacks the Fourier-support separation used in the Besov case: the text says 'One can mimic the above proof' without proving that the limiting solution is nonzero, and Corollary 1.6 is stated without a proof. These are genuine correctness gaps, but they are not circular reductions: no equation is equal to an input by construction, and no fitted parameter is relabeled as a prediction.

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

No free parameters are fitted to data; the auxiliary convex-integration parameters λ, μ, γ, and σ are chosen inside the proof and do not enter the statements. No new physical entities are postulated. The main modeling input is the paraproduct tensor-product definition of the nonlinear term.

assumptions (6)

domain assumption Paraproduct definition of u⊗u in H^{-s} for distributions (Eqs (1.3)–(1.5), Definition 1.3).
The entire notion of singular mild and stationary solution depends on this extension, since u need not be in L^2_loc and u⊗u need not be in L^1. The theorems are about this extended solution class.
standard math Littlewood-Paley/Besov theory and Bernstein inequalities (Section 1.3 and throughout).
Definitions and frequency-block estimates are used in all propositions and theorems.
standard math Heat semigroup smoothing and Oseen-kernel estimates (Lemma 2.4, cited to [MYZ08]).
Used for local well-posedness and the fixed-point proof of Proposition 2.1.
standard math Mikado-flow building blocks and estimates (Lemma 6.1, via [DS17, DLS13, CL22]).
Core geometric/convex-integration input; assumed as prior proved theorems.
standard math Unconditional uniqueness of the linear heat equation in the distribution class (proof of Proposition 1.4).
Needed to identify constant singular solutions with mild solutions.
standard math Improved Hölder inequality (used in Lemma 9.1, cited to [MS18, BV19b]).
Supports the L^2 oscillation estimate in the L^2-iteration.

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Pith. "Pith review of On non-uniqueness of mild solutions and stationary singular solutions to the Navier-Stokes equations." pith.science (2026). https://pith.science/paper/YYEA3UG3

@misc{pith2026260303666,
  author       = {Pith},
  title        = {Pith review of: On non-uniqueness of mild solutions and stationary singular solutions to the Navier-Stokes equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YYEA3UG3}},
  note         = {Machine review of arXiv:2603.03666}
}

read the original abstract

We prove that the unconditional uniqueness of mild solutions to the Navier-Stokes equations fails in all the Besov spaces with negative regularity index, by constructing non-trivial stationary singular solutions via convex integration. We also establish uniqueness of stationary weak solutions in an endpoint critical space. Similar results are proved for the fractional Navier-Stokes equations with arbitrarily large power of the Laplacian in both Lebesgue and Besov spaces.

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4 extracted references · 2 linked inside Pith · cited by 1 Pith paper

[2]

[Che92] J.-Y

arXiv:2511.09556. [Che92] J.-Y. Chemin. Remarques sur l’existence globale pour le syst` eme de Navier-Stokes incompressible.SIAM J. Math. Anal., 23(1):20–28,

arXiv
[2019]

[Tri83] H

https://terrytao.wordpress.com/2019/01/08/255b- notes-2-onsagers-conjecture/. [Tri83] H. Triebel.Theory of function spaces, volume 78 ofMono- graphs in Mathematics. Birkh¨ auser Verlag, Basel,

2019
[2025]

[BCD11] H

arXiv:2506.00841. [BCD11] H. Bahouri, J.-Y. Chemin, and R. Danchin.Fourier analy- sis and nonlinear partial differential equations, volume 343 ofGrundlehren der mathematischen Wissenschaften [Fun- damental Principles of Mathematical Sciences]. Springer, Heidelberg,

arXiv
[2026]

[Kat84] T

arXiv:2602.19846. [Kat84] T. Kato. StrongL p-solutions of the Navier-Stokes equa- tion inR m, with applications to weak solutions.Math. Z., 187(4):471–480,

arXiv

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [2]

[Che92] J.-Y

arXiv:2511.09556. [Che92] J.-Y. Chemin. Remarques sur l’existence globale pour le syst` eme de Navier-Stokes incompressible.SIAM J. Math. Anal., 23(1):20–28,

arXiv

[2] [2019]

[Tri83] H

https://terrytao.wordpress.com/2019/01/08/255b- notes-2-onsagers-conjecture/. [Tri83] H. Triebel.Theory of function spaces, volume 78 ofMono- graphs in Mathematics. Birkh¨ auser Verlag, Basel,

2019

[3] [2025]

[BCD11] H

arXiv:2506.00841. [BCD11] H. Bahouri, J.-Y. Chemin, and R. Danchin.Fourier analy- sis and nonlinear partial differential equations, volume 343 ofGrundlehren der mathematischen Wissenschaften [Fun- damental Principles of Mathematical Sciences]. Springer, Heidelberg,

arXiv

[4] [2026]

[Kat84] T

arXiv:2602.19846. [Kat84] T. Kato. StrongL p-solutions of the Navier-Stokes equa- tion inR m, with applications to weak solutions.Math. Z., 187(4):471–480,

arXiv