Nonuniqueness analysis on the Navier-Stokes equation in $C_{t}L^{q}$ space

Changxing Miao; Zhiwen Zhao

REVIEW 3 major objections 3 minor 1 cited by

Nonuniqueness analysis on the Navier-Stokes equation in $C_{t}L^{q}$ space

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Pith's one-line read The paper proves that for a fixed exponent $q$ slightly above $2$, the three-dimensional Navier-Stokes equations admit weak solutions, continuous in time with values in $L^q$, that realize any prescribed smooth nonnegative kinetic energy…

desk verdict A fresh L^q-normalized jet idea, but a load-bearing arithmetic error at (3.22) makes the main theorem unsupported. read the letter →

arxiv 2501.09698 v3 pith:EPSJEVQE submitted 2025-01-16 math.AP

Changxing Miao , Zhiwen Zhao This is my paper · ORCID

classification math.AP MSC 35Q3076D0535D30

keywords Navier-StokesnonuniquenessweaksolutionsintermittentconvexintegrationLq-normalizedjetsprescribedkineticenergyC_tL^qzeroinitialdata

topics Navier-Stokes Regularity

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 aims to prove that the three-dimensional Navier-Stokes equations admit far more weak solutions than classical well-posedness theorems suggest: there is a single integrability exponent $q$ slightly above $2$ such that, for any prescribed smooth nonnegative kinetic energy profile $e(t)$, one can find a weak solution $u\in C_t([0,T];L^q(\mathbb{T}^3))$ whose squared $L^2$ norm equals $e(t)$ at every time. This matters because it turns the kinetic energy from a quantity that might identify a unique evolution into a freely assignable datum, producing infinitely many distinct solutions all starting from rest. The novelty is that the construction works directly in $L^q$: the building blocks are intermittent jets normalized in $L^q$ rather than $L^2$, so no interpolation inequality is needed, and the solutions inherit a mild fractional spatial regularity $C_t W^{\alpha,q}$ for some $0<\alpha\ll1$. If correct, the result sharply locates where nonuniqueness begins on the integrability scale relative to the critical exponent $q=3$.

What carries the argument

The central object is the $L^q$-normalized intermittent jet $$ $W^{{(\zeta)}}$(x,t)=\psi_r(N_\Lambda\$\lambda$\$\sigma$(x\cdot\zeta+\mu t))\;\varphi_\$\sigma$\big(N_\Lambda\$\lambda$\$\sigma$(x-\alpha_\zeta)\cdot A_\zeta,\,N_\Lambda\$\lambda$\$\sigma$(x-\alpha_\zeta)\cdot(\zeta\times A_\zeta)\big)\,\zeta, $$ a vector-valued building block with support concentrated on a small set and oscillations at frequency $\lambda$; it satisfies $\int|W^{(\zeta)}|^q\,dx=1$. The proof uses these jets, rather than $L^2$-normalized Beltrami waves, because they have disjoint supports and carry the $L^q$ normalization explicitly. Their averaged tensor products obey the decomposition identity (3.8), which represents every small symmetric matrix $R$ as a positive combination of $-\int W^{(\zeta)}\otimes W^{(\zeta)}\,dx$; this identity is what cancels the previous Reynolds stress in each step. Around this core the iteration adds a time corrector to remove the time derivative of the principal part, an incompressibility corrector to make the velocity divergence-free, and a partition-of-unity energy mechanism that forces the squared $L^2$ norm to follow $e(t)$, including at its zeros.

What would settle it

Check whether the admissible set defined by (2.7), (3.23), (3.25), (3.28) and (3.34) is nonempty: producing a single explicit numerical tuple $(q,\varepsilon,\varepsilon_*,A,a,b,\beta,p)$ that meets all these constraints would validate the consistency of the iteration, while a proof that no such tuple exists would directly falsify Proposition 2.1 and with it Theorem 1.2.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.2: there exists a uniform exponent $2<q\ll3$ such that for any nonnegative smooth function $e(t):[0,T]\to[0,\infty)$ there is a weak solution $u\in C_t([0,T];L^q(\mathbb{T}^3))$ of the periodic Navier-Stokes system with $\int_{\mathbb{T}^3}|u(x,t)|^2\,dx=e(t)$ for every $t$. Taking $e_k(t)=1-\cos(kt)$ for integers $k\ge1$ yields infinitely many distinct nontrivial weak solutions all emanating from zero initial data. The proof is constructive: an inductive scheme produces a sequence of smooth solutions of the Navier-Stokes-Reynolds system whose Reynolds stresses decay superexponentially, and the limit solves the true Navier-Stokes equation while tracking the assigned energy. By the fractional Gagliardo-Nirenberg inequalities the constructed solution is further shown to lie in $C_t W^{\alpha,q}$ for some $0<\alpha\ll1$, upgrading the regularity obtained in earlier $L^2$-normalized constructions.

Load-bearing premise

The proof rests on the joint existence of parameters $(q,\varepsilon,\varepsilon_*,A,a,b,\beta,p)$ satisfying the constraints (2.7), (3.23), (3.25), (3.28), (3.34), together with the auxiliary inequalities in Lemma 3.12; the paper asserts such parameters exist but does not give an explicit tuple, and if the admissible set is empty the inductive construction collapses.

Editorial extensions

If this is right

Every smooth nonnegative kinetic energy profile $e(t)$ is realized by at least one weak solution in $C_tL^q$; in particular $e_k(t)=1-\cos(kt)$ yields infinitely many distinct nontrivial solutions starting from zero initial data.
The exponent $q$ is fixed once and for all, independent of the chosen profile $e(t)$, so the nonuniqueness is uniform across all energy histories.
The constructed solutions belong to $C_tW^{\alpha,q}$ for some $0<\alpha\ll1$, a spatial-regularity upgrade over earlier $C_tH^\alpha$ constructions.
The scheme keeps the $L^q$ bound direct, without interpolation inequalities, which the paper proposes as a stepping stone for future convex-integration constructions.

Reading between the lines

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

It would be worth testing numerically whether an explicit admissible tuple $(q,\varepsilon,\varepsilon_*,A,a,b,\beta,p)$ satisfying all of (2.7), (3.23), (3.25), (3.28) and (3.34) exists; the paper asserts existence but does not display one, and a concrete example would make the iteration computationally checkable.
The same $L^q$-normalized jet construction could plausibly be transplanted to other convex-integration settings, such as transport equations or the Euler equations, where direct $L^q$ control without interpolation might shorten existing arguments.
If the theorem is correct, a consequence the authors do not spell out is an extreme form of nonuniqueness at zero initial data: the kinetic energy alone, at this regularity level, does not select a unique evolution.

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Formalized claims in Lean

Claim #1: On the paper's own terms, the central discovery is Theorem 1.2: there exists a uniform exponent $2<q\ll3$ such that for any nonnegative smooth function $e(t):[0,T]\to[0,\infty)$ there is a weak solution $u\in C_t([0,T];L^q(\mathbb{T}^3))$ of the periodic Navier-Stokes system with $\int_{\mathbb{T}^3}|u(x,t)|^2\,dx=e(t)$ for every $t$. Taking $e_k(t)=1-\cos(kt)$ for integers $k\ge1$ yields infinite

/-- @claim 1 On the paper's own terms, the central discovery is Theorem 1.2: there exists a uniform exponent $2<q\ll3$ such that for any nonnegative smooth function $e(t):[0,T]\to[0,\infty)$ there is a weak solution $u\in C_t([0,T];L^q(\mathbb{T}^3))$ of the periodic Navier-Stokes system with $\int_{\mathbb{T}^3}|u(x,t)|^2\,dx=e(t)$ for every $t$. Taking $e_k(t)=1-\cos(kt)$ for integers $k\ge1$ yields infinite -/
def central_claim : Prop :=

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

The L^q-normalized intermittent jets are a genuinely new ingredient, and the paper is carefully organized around the Buckmaster-Vicol machinery. The bad news: the central scale-separation estimate (3.22) is wrong, and the error is load-bearing. As written, the exponent in (3.22) is always positive for every admissible parameter range, so ℓ^{-10}(λσ)^{-1/q} diverges as m→∞ instead of being ≪1. That estimate is what controls the error term from the improved Hölder inequality in the proof of Proposition 3.6, so the perturbation bound fails and the induction collapses. The reader's worry about a missing parameter tuple undersells the problem: the displayed algebra makes the admissible set empty.

What's worth keeping: the L^q-normalized jets and the direct construction in C_t L^q without interpolation are real steps forward in principle, and the energy iteration with prescribed kinetic energy is well thought out. The paper is honest about being inside the Buckmaster-Vicol program, and the parameter discussion shows the authors understand where the threshold q=3 is supposed to enter.

The soft spots beyond the main error: Remark 1.3 about W^{α,q} is asserted via a reference without a derivation, and the abstract's language about q arbitrarily close to 3 is not supported by the proof, which keeps q near 2. The stress-test note's side claim about τ2 being negative for all admissible b is not needed and may be wrong; the core arithmetic stands on its own.

This paper is for convex-integration specialists. The idea might be salvageable with a different choice of ℓ or a different jet scaling, but the current proof does not establish Theorem 1.2. I would send it to a serious referee for a careful check of the scaling and parameter constraints; the result is important enough to merit that attention, and the flaw is local rather than conceptual.

Referee Report

3 major / 3 minor

Summary. The paper proposes an intermittent convex integration construction of weak solutions to the three-dimensional Navier-Stokes equations in C_t L^q for a uniform exponent 2<q<<3, with arbitrarily prescribed kinetic energy and zero initial data, and without the use of interpolation inequalities. The main iterative statement is Proposition 2.1, which asserts an inductive step for the Navier-Stokes-Reynolds system with the estimates (2.4)-(2.7); Theorem 1.2 is then inferred by passing to the limit. The construction introduces L^q-normalized intermittent jets in Section 3.2, builds a perturbation w_{m+1} in Section 3.3, estimates it in Proposition 3.6, bounds the new Reynolds stress in Lemma 3.10, and closes the energy iteration in Lemma 3.12. The proof follows the standard convex-integration pattern and does not rely on fitted parameters, but the central smallness condition in Proposition 3.6 is arithmetically impossible.

Significance. Conditional significance is high. If Theorem 1.2 were correct, it would provide nonunique C_t L^q weak solutions with q>2, prescribed kinetic energy, and zero initial data, going beyond interpolation-based arguments and moving toward the L^3 critical threshold; the claimed C_t W^{alpha,q} improvement would also be noteworthy. The paper has real strengths: the parameters are governed by explicit inequalities, the prescribed energy is an input rather than a fitted quantity, and the overall scheme follows the established Buckmaster-Colombo-Vicol strategy with L^q-normalized jets. However, the decisive estimate (3.22) has an impossible exponent, and this cannot be repaired by a particular choice of parameters, so the claimed result is not supported.

major comments (3)

[Section 3.4, Eq. (3.22) and Proposition 3.6] The decisive smallness condition in (3.22) is arithmetically false. From (3.1), delta_{m+1}^{1/2}/delta_m^{1/2}=b^{-beta}, theta_{m+1}^{(q-2)/q}=lambda_{m+1}^{-(q-2)(1+epsilon*)}, and lambda_m=lambda_{m+1}/b, one obtains ell = lambda_{m+1}^{-5-(q-2)(1+epsilon*)} b^{5-beta}. Hence ell^{-10}(lambda_{m+1}sigma)^{-1/q} = lambda_{m+1}^{10[5+(q-2)(1+epsilon*)] - (3-q-(q+2)epsilon*)/(3q)} b^{-10(5-beta)}. For all admissible q in (2,3) and epsilon*<1/4, the lambda-exponent is at least 50 - 1/6 >0, so the quantity tends to infinity as m grows and can never be much smaller than 1. Even taking the manuscript's displayed exponent in (3.22) at face value, the extra positive term 10 beta(b-1)/b only makes the exponent larger. Therefore the estimate leading to ||w_{m+1}^{(p)}||_{L^q} <= (2/3) kappa_* delta_{m+1}^{1/2} in (3.26) is invalid, and with it Proposition 3.6, Corollary 3.8, Lemma 3.10, Proposition 2.1, and Theorem 1.2 collapse.
[Section 3.4, Eq. (3.23)] The parameter condition (3.23) derived from (3.22) is equally impossible. The desired inequality 10[(q-2)(1+epsilon*)+5+beta(b-1)/b] <= (3-q-(q+2)epsilon*)/(6q) has left-hand side at least 50 while the right-hand side is at most 1/12 for q>2 and 3-q-(q+2)epsilon*<1. No choice of b can satisfy this inequality, and the accompanying lower bound on b does not remove the leading term 5 in the left-hand side. Thus the constraint set used to close the L^q estimate is empty.
[Section 3.5, Lemma 3.10; Section 2, Proposition 2.1] Independently of the failure of (3.22), the manuscript does not establish joint feasibility of the parameter constraints (2.7), (3.23), (3.25), (3.28), and (3.34). For example, (3.34) forces b >= 720/(epsilon*-2epsilon) and q <= 2 + (epsilon*-2epsilon)/(4(b+35)(1+epsilon*)), but no explicit tuple is exhibited and no monotonicity argument is given to prove that all constraints can hold simultaneously. Since the inductive step depends on these inequalities to control every term in the Reynolds stress estimate, this is a load-bearing gap in the proof as written.

minor comments (3)

[Section 2, proof of Proposition 2.1] The heading 'Proof of Theorem 2.1' appears to be a typo; the proof establishes Proposition 2.1 and then Theorem 1.2.
[Lemma 3.5] The notation 'p√11' should be typeset as 11^{1/p} or \sqrt[p]{11}; as printed it is confusing.
[Section 3.2] The function Phi is introduced as a map R^2 -> R^2, but later Phi_sigma and phi_sigma are used as scalar-valued functions in the definitions of the jets; the domain and codomain should be made consistent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof is a self-contained convex-integration construction with the prescribed energy as an input.

full rationale

The paper's derivation chain is a standard intermittent convex-integration iteration: Theorem 1.2 is reduced to Proposition 2.1, which is proven by constructing u_{m+1} from u_m, the Reynolds stress R_m, and the prescribed kinetic energy e(t). The prescribed function e(t) is a theorem input, not a fitted parameter, and the energy identity (3.38) is an exact algebraic consequence of the choice of coefficients a(ζ) and the geometric decomposition of Lemma 3.4. The constants q, ε, ε*, a, b, β, p are chosen to satisfy explicit inequalities (2.7), (3.23), (3.25), (3.28), (3.34), and the auxiliary estimates in Lemma 3.12. No step of the proof assumes Theorem 1.2 or Proposition 2.1 itself. The cited results — Buckmaster–Vicol building blocks and estimates, Modena–Szekelyhidi improved Hölder inequality, De Lellis–Székelyhidi antidivergence operator, and the fractional Gagliardo–Nirenberg inequalities — are external and are used as tools, not as a substitute for the present construction. Possible arithmetic impossibility of the parameter constraints would be a correctness flaw, not circularity. Therefore the appropriate finding is no significant circularity.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The construction introduces no new physical entities; the free parameters q, ε, ε*, A, a, b, β and p are chosen by the authors to satisfy explicit inequalities. The main load-bearing input beyond standard convex integration lemmas is the joint feasibility of these parameter constraints, which is asserted but not demonstrated by an explicit tuple.

free parameters (8)

q
Target Lebesgue exponent chosen in (2,3), ultimately close to 2, satisfying the parameter constraints in (3.23)-(3.34).
ε
Regularity slack parameter with 0<2ε<ε*<1/4, used to control derivative losses in (2.4) and (3.28).
ε*
Accuracy parameter defined as (3-q)/A in (2.3), with A>q+2, and required to satisfy 3-q-(q+2)ε*>0.
A
Free parameter larger than q+2 used to define ε* in (2.3).
a
Frequency base in λ_m=a^{b^m}; chosen large enough and satisfying the integrality condition (2.7); depends on sup e(t) according to Remark 2.2.
b
Frequency exponent factor, chosen integer and large to satisfy (3.23), (3.25), (3.28), (3.34).
β
Decay exponent in δ_m; chosen positive and small as required in (3.34) and Lemma 3.12.
p
Auxiliary integrability exponent in (1,2), selected in (3.34) for the Reynolds stress estimate Lemma 3.10.

assumptions (6)

standard math Symmetric positive matrices near identity can be decomposed as a finite sum of rank-one pieces γ(ζ)^2 ζ⊗ζ
Invoked as Lemma 3.4 to represent Id - R/ρ_i and to build the principal perturbation; cited to [20,64].
domain assumption L^q-normalized intermittent jets W^(ζ) satisfy the normalization, disjoint-support and derivative estimates in (3.7)-(3.9)
The entire iteration relies on these scaling identities and on disjoint supports of the jets; periodicity requires the arithmetic condition (2.7).
standard math Improved Hölder inequality of Lemma 3.7
Used in Proposition 3.6 to bound products of high-frequency waves with slowly varying coefficients; cited to [51].
standard math Frequency projection estimate of Lemma 3.11
Used to estimate the high-frequency part of the oscillation error; cited to [11].
standard math Fractional Gagliardo-Nirenberg inequalities from [6]
Invoked in Remark 1.3 for the claimed C_t W^{α,q} improvement.
ad hoc to paper Joint feasibility of the parameter constraints (2.7), (3.23), (3.25), (3.28), (3.34)
The paper asserts existence of a parameter tuple without giving an explicit example; this is load-bearing for Proposition 2.1.

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Pith. "Pith review of Nonuniqueness analysis on the Navier-Stokes equation in $C_{t}L^{q}$ space." pith.science (2026). https://pith.science/paper/EPSJEVQE

@misc{pith2026250109698,
  author       = {Pith},
  title        = {Pith review of: Nonuniqueness analysis on the Navier-Stokes equation in $C_tL^q$ space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EPSJEVQE}},
  note         = {Machine review of arXiv:2501.09698}
}

abstract

In the presence of any prescribed kinetic energy, we implement the intermittent convex integration scheme with $L^{q}$-normalized intermittent jets to give a direct proof for the existence of solution to the Navier-Stokes equation in $C_{t}L^{q}$ for some uniform $2<q\ll3$ without the help of interpolation inequality. The result shows the sharp nonuniqueness that there evolve infinite nontrivial weak solutions of the Navier-Stokes equation starting from zero initial data. Furthermore, we improve the regularity of solution to be of $C_{t}W^{\alpha,q}$ in virtue of the fractional Gagliardo-Nirenberg inequalities with some $0<\alpha\ll1$. More importantly, the proof framework provides a stepping stone for future progress on the method of intermittent convex integration due to the fact that $L^{q}$-normalized building blocks carry the threshold effect of the exponent $q$ arbitrarily close to the critical value $3$.

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Forward citations

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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