Non-uniqueness of Leray--Hopf solutions for the $3D$ fractional Navier--Stokes equations perturbed by transport noise

Andre Schenke; Marco Rehmeier; Theresa Lange

Non-uniqueness of Leray--Hopf solutions for the $3D$ fractional Navier--Stokes equations perturbed by transport noise

T0 review · 0 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proves that transport noise does not restore uniqueness: for very small fractional diffusion exponents, infinitely many Leray–Hopf solutions can start from the same deterministic velocity field.

desk verdict First Leray–Hopf non-uniqueness for unforced stochastic fractional NSE with transport noise: a serious, detailed proof whose main novelty holds up, though the admissible exponent is minuscule and the energy inequality is local in time. read the letter →

arxiv 2412.16532 v1 pith:P4I5JIPS submitted 2024-12-21 math.AP math-phmath.MPmath.PR

Theresa Lange , Marco Rehmeier , Andre Schenke This is my paper · ORCID

classification math.APmath-phmath.MPmath.PR MSC 60H1535Q3035A02

keywords fractionalNavier–StokesequationsLeray–Hopfsolutionstransportnoisenon-uniquenessconvexintegrationstochasticpartialdifferentialflowtransformationBesovspaces

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

This paper proves that adding transport noise to the 3D fractional Navier–Stokes equations does not restore uniqueness: for fractional diffusion exponents $\alpha<\alpha_0\approx 8.7\times 10^{-9}$, it constructs infinitely many Leray–Hopf solutions starting from the same deterministic initial velocity. The solutions are global in time, probabilistically strong, and solve the equation with no additional forcing term. They satisfy the pathwise energy inequality up to a random stopping time, yet any two of them differ on that interval almost surely. This matters because it supplies the first Leray–Hopf non-uniqueness result for the unforced fractional Navier–Stokes equations under any stochastic perturbation, closing a gap left by earlier constructions that needed a specially chosen force.

What carries the argument

The proof uses a flow transformation: the stochastic flow $\Phi$ of the Stratonovich SDE $d\Phi=\sum_k\sigma_k(\Phi)\circ dB_k$ conjugates the SPDE (1.1) into a PDE with random coefficients (2.5), involving flowed operators $\mathrm{div}_\Phi$, $\nabla_\Phi$, and $(-\Delta)^\alpha_\Phi$; solutions are mapped back by $u(t)=v(t)\circ\Phi(t)^{-1}$, preserving kinetic energy and regularity. On the transformed equation, a pathwise convex-integration scheme iterates over modified Beltrami waves with an energy-pumping term steering the velocity toward a prescribed energy profile. The genuinely new fractional contributions produce additional flow and mollification errors, controlled by a Besov interpolation lemma (Lemma 8.3) whose dimension-dependent constant satisfies $C_3(L,s,1)\le C L^8$. This estimate, together with the imported stochastic-flow bounds (4.3)–(4.5), is what forces the extremely small admissible range $\alpha<\alpha_0\approx 8.7\times 10^{-9}$.

What would settle it

Evaluate the norm of the operator $T(h)=h\circ\psi^{-1}-h$ in Lemma 8.3 for $d=3$, $s=\delta+2\alpha-1$: if the constant grows like $L^{8+\varepsilon}$ for any $\varepsilon>0$, the flow-error estimate in Section 8.7.2 cannot be absorbed into $\delta_{n+2}$ and the construction fails. Equivalently, a direct check of the Wong–Zakai approximation bound (4.3) on $\mathbb{T}^3$ would test the same load-bearing estimate.

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Extended reading notes

Core claim

Theorem 2.4 states that for every $0<\alpha<\alpha_0:=1/(2cb+1)$, with $b=38$ and $c$ as in Section 4.2 (numerically $\alpha_0\approx 8.7\times 10^{-9}$), there exists a deterministic initial velocity $u_0\in L^2(\mathbb{T}^3)$, an almost surely strictly positive stopping time $\tau_0$, and infinitely many $\tau_0$-Leray–Hopf solutions to the stochastic fractional Navier–Stokes system (1.1) with initial condition $u_0$ and paths in $C(\mathbb{R}_+,C^\theta(\mathbb{T}^3))$ for some $\theta>\alpha$. Any two of these solutions are distinct on $[0,\tau_0]$ almost surely. The statement is new because the equation carries no deterministic forcing term, and it constitutes the first Leray–Hopf non-uniqueness result for the unforced fractional Navier–Stokes equations with any stochastic perturbation. The solutions are global in time and satisfy the pathwise energy inequality on $[0,\tau_0]$.

Load-bearing premise

Everything rests on the imported quantitative stochastic-flow estimates (4.3)–(4.5) and on the Besov interpolation bound $C_3(L,s,1)\le C L^8$ of Lemma 8.3; if either fails, the flow-error and mollification-error controls in the convex-integration iteration collapse.

Editorial extensions

If this is right

For every sufficiently small diffusion exponent $\alpha$, the class of $\tau_0$-Leray–Hopf solutions is not a uniqueness class: one deterministic $L^2$ initial condition admits infinitely many distinct solutions.
The pathwise energy inequality holds on the non-empty random interval $[0,\tau_0]$ for all constructed solutions, so the non-uniqueness persists inside the physically relevant Leray–Hopf subclass.
Restricting to weakly dissipative, transport-noised equations does not eliminate the non-uniqueness phenomenon even when the forcing term used in earlier constructions is removed.
For the wider range $\alpha<\tilde\alpha_0\approx 1.7\times 10^{-4}$, the same iteration still produces global solutions with prescribed energy profiles, although without the energy inequality.

Reading between the lines

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

Beyond the paper: if the Besov interpolation constant in Lemma 8.3 can be sharpened, the admissible range of $\alpha$ should grow well beyond $10^{-8}$, potentially toward the deterministic threshold $\alpha<1/3$ known in the unforced case.
Beyond the paper: the same flow-transform and convex-integration template is a natural candidate for other dissipative SPDEs with transport noise, provided the analogous flow and mollification errors can be balanced with a milder interpolation loss.
Beyond the paper: a direct numerical check of the Wong–Zakai approximations $\varphi_n$ against the bound (4.3) on the torus would test the quantitative core of the stochastic-flow step before any further theoretical refinement.

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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

This is a real result: the first Leray–Hopf non-uniqueness theorem for the unforced fractional Navier–Stokes equations with transport noise. It removes the deterministic forcing that all earlier stochastic results needed, and it does so via a long, careful convex integration argument that tracks the energy profile through every constant. That is substantial work, and the authors are honest about the limitations—the admissible fractional exponent is about 8.7×10^-9, the energy inequality only holds up to a random stopping time, and the initial datum cannot be prescribed.

The stress-test flagged Lemma 8.3 as a potential fatal gap. I read that lemma and the surrounding proof carefully, and I think the concern is misplaced. The p=∞ endpoint of the interpolation does not apply Lemma B.8 to a negative-regularity Besov space; because s is negative, −s−ε/4 is positive, and the lemma's composition estimate for positive Hölder norms is exactly what is needed. The displayed power of L is consistent with applying the flow bound twice. So the central load-bearing estimate looks acceptable.

The soft spots are real but not fatal. The proof relies heavily on imported estimates from [23], and while the authors state them clearly, a referee will need to verify the flow estimates and the constant tracking independently. The parameter choices in Section 7.4 form a delicate web; a small error in the Besov constants would propagate to the admissible range of α, and the current range is already tiny. The result is also local in time for the energy inequality, so the global non-uniqueness question remains open. These are limitations the authors themselves acknowledge, not hidden defects.

Who gets value from this: researchers in stochastic PDEs, convex integration, and the ill-posedness of fluid equations. It is not a paper to assign whole in a reading group, but the main theorem and Lemma 8.3 would be worth discussing. I would not desk-reject this. It deserves a serious referee who checks the imported estimates and the Besov interpolation, even if heavy revisions are expected.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that for each sufficiently small α > 0 (specifically α < α0 ≈ 8.7·10^-9), the 3D fractional Navier–Stokes equations on the torus perturbed by Stratonovich transport noise admit infinitely many probabilistically strong, analytically weak Leray–Hopf solutions starting from the same deterministic L2 initial velocity, with paths in C(R+, C^θ) for some θ > α, and distinct on a common strictly positive random time interval. The proof combines a flow transformation that rewrites the SPDE as a PDE with random coefficients, a convex integration scheme adapted from Hofmanová–Lange–Pappalettera [23], and a new Besov interpolation estimate (Lemma 8.3) that controls the error terms arising from the interaction of the fractional Laplacian with the flow. The paper carefully tracks all energy-profile-dependent constants so that the pathwise energy inequality can be closed.

Significance. If correct, this is the first Leray–Hopf non-uniqueness result for the unforced stochastic fractional Navier–Stokes equations, and it also improves the existence theory for analytically weak solutions to (1.1). The proof is exceptionally detailed: the main iterative proposition includes explicit parameter choices, the energy profile dependence is tracked through every constant, and the new Besov interpolation lemma is stated and proved with an explicit constant C_3(L,s,1) ≤ C L^8. The admissible range α < α0 is very small, but the authors are transparent about this limitation. The paper also provides falsifiable predictions in the sense that the constructed solutions have prescribed energy profiles. The main new technical step, Lemma 8.3, is internally consistent; the potential concern about its p=∞ endpoint does not land because the relevant Besov norm has positive regularity.

minor comments (3)

[Section 8.7.2, proof of Lemma 8.3] The p=∞ endpoint of Lemma 8.3 cites Lemma B.8 tersely; since the norm being bounded is B^{-s-ε/4}_{∞,∞} with -s-ε/4 ∈ (0,1), the identification with the classical Hölder space via (1.4) should be stated explicitly so that the positive-regularity application of Lemma B.8 is evident.
[Equation (4.16) and Section 7.6] The displayed expression for α0 in (4.16) appears to contain a typo in the denominator; it should match the formula 1/(2bc+1) = [m − 1]/[2(1+ε)(m+ε)^5 + m − 2 − ε] given in Section 7.6.
[Section 6.1, parameter choices] The relations listed after (6.4)–(6.9) are asserted to follow from the definitions, but a short derivation of the most delicate one, (6.8), would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity: the convex-integration construction prescribes, rather than fits, the energy profiles, and the cited [23] estimates are external auxiliary support despite author overlap.

full rationale

The paper's central claim is a pathwise convex-integration existence/non-uniqueness theorem. The energy profiles in Proposition 3.3 are prescribed as inputs, not fitted to data, and the iterative Proposition 4.1 proves existence of solutions realizing those profiles. Non-uniqueness then follows legitimately from profiles that agree at time zero but differ on a sequence tending to zero (Proposition 3.3(ii)); this is a standard witness construction, not a prediction equivalent to its input. The theorem is not a renaming of a known result: it is the first unforced Leray--Hopf non-uniqueness statement for the stochastic fractional Navier--Stokes equations, and it extends the deterministic constructions of [12,15] to the transport-noise setting via the flow transformation. The paper does rely heavily on the prior work [23], which overlaps with author T. Lange, for stochastic-flow estimates (4.3)--(4.5) and several Besov lemmas. This is a normal and substantial self-citation, but the cited results are published, auxiliary, assumption-based estimates; they are not the statement of Theorem 2.4, and no equation in the paper reduces the conclusion to them by construction. A skeptical concern that Lemma 8.3's p=infinity endpoint invokes the positive-Holder composition estimate Lemma B.8 for a negative Besov norm is best classified as a proof-gap or correctness risk at that step, not as circularity. Overall, no load-bearing step turns the claimed derivation into its own input, so the appropriate finding is no significant circularity; the score reflects only the acknowledged heavy overlap with [23].

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

No new physical entities, particles, or forces are introduced. The modified Beltrami waves, flow-transported operators, and stopping times are mathematical construction tools with no independent physical evidence requirement. The free parameters are all internal to the convex integration scheme or to the engineered energy profiles, and the main axioms are standard stochastic-flow and Besov-space tools, plus an ad hoc energy-profile choice.

free parameters (5)

fractional exponent α = 0 < α < α0 ≈ 8.7e-9
The theorem is restricted to α below the constant α0 = 1/(2cb+1) derived in Section 7.4; this is not a physical input but a direct consequence of the estimates, especially (7.22).
convex integration base parameter a = a = (A(1+¯e/e)(¯e/e)^{2r})^{1/(ε+1/2)}, see (7.23)
a is chosen to satisfy six lower bounds a1 through a6 so that all energy-dependent constants can be absorbed; it is a hand-chosen construction parameter.
constant Mv = unique solution of E = Mv/(1+Mv^{2δ}), see (7.24)
Mv controls the velocity increment estimate (4.12) and is determined by a fixed-point equation depending on the energy profile; it is a construction parameter.
exponential parameters m, ε, b, c = m=23, ε=15, b=38, c=(b^4(1+ε)-1/2)/(b-1-ε)
These exponents are chosen in Section 7.3 to make all powers of L and δ close in the iterative estimates; they are ad hoc to the convex integration scheme and determine the tiny value of α0.
energy profiles eε_k = family in E with common initial value, inf ε/2, sup ε, slope ≤ -1/2 on [0,r)
The non-uniqueness is engineered by choosing distinct energy profiles with the same initial value; the initial condition of the constructed solutions is an output, not a prescribed datum.

assumptions (4)

domain assumption Quantitative flow approximation estimates (4.3)-(4.5) for mollified flows φ_n, including Wong-Zakai convergence and rough-path localization with stopping times t_L.
Imported from [23, Lemmas 2.1-2.2], these bounds control all differences between φ_n and φ and are used throughout the convex integration scheme, especially in Sections 6.7.3 and 8.7.
standard math Besov composition and negative-regularity lemmas, in particular Lemma 8.3 with constant C3(L,s,1) ≤ C L^8, and Lemmas B.1-B.15.
These lemmas are the technical backbone for the new flow and mollification error terms; they are cited from [4,23,38] and stated in Appendix B but not all are proved in this paper.
domain assumption The SPDE-to-PDE equivalence via the measure-preserving stochastic flow Φ, including C∞ diffeomorphism regularity and the semimartingale identity in Lemma 2.8.
Standard from Kunita [31], this equivalence converts the stochastic equation (1.1) into the random PDE (2.5) and is the foundation of the entire proof.
ad hoc to paper The chosen energy profiles eε_k have common initial value, continuous differentiability, and strict decrease with slope ≤ -1/2 on [0,r).
The strict decrease converts the small Hölder bound (3.2) into the pathwise energy inequality in Step 3 of Section 3; this is a constructed condition, not a physical input.

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Pith. "Pith review of Non-uniqueness of Leray--Hopf solutions for the $3D$ fractional Navier--Stokes equations perturbed by transport noise." pith.science (2026). https://pith.science/paper/P4I5JIPS

@misc{pith2026241216532,
  author       = {Pith},
  title        = {Pith review of: Non-uniqueness of Leray--Hopf solutions for the $3D$ fractional Navier--Stokes equations perturbed by transport noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P4I5JIPS}},
  note         = {Machine review of arXiv:2412.16532}
}

abstract

For the $3D$ fractional Navier--Stokes equations perturbed by transport noise, we prove the existence of infinitely many H\"older continuous analytically weak, probabilistically strong Leray--Hopf solutions starting from the same deterministic initial velocity field. Our solutions are global in time and satisfy the energy inequality pathwise on a non-empty random interval $[0,\tau]$. In contrast to recent related results, we do not consider an additional deterministic suitably chosen force $f$ in the equation. In this unforced regime, we prove the first result of Leray--Hopf nonuniqueness for fractional Navier--Stokes equations with any kind of stochastic perturbation. Our proof relies on convex integration techniques and a flow transformation by which we reformulate the SPDE as a PDE with random coefficients.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [23]

Hofmanov´ a, T

M. Hofmanov´ a, T. Lange, and U. Pappalettera. Global existe nce and non-uniqueness of 3D Euler equations perturbed by transport noise. Probability Theory and Related Fields , 2023

work page 2023

[2] [1]

Albritton, E

D. Albritton, E. Bru´ e, and M. Colombo. Non-uniqueness of Lera y solutions of the forced Navier- Stokes equations. Ann. of Math. (2) , 196(1):415–455, 2022. 94

work page 2022

[3] [2]

Albritton, E

D. Albritton, E. Bru´ e, and M. Colombo. Gluing non-unique Navier- Stokes solutions. Ann. PDE , 9(2):Paper No. 17, 25, 2023

work page 2023

[4] [3]

Albritton and M

D. Albritton and M. Colombo. Non-uniqueness of Leray Solutions t o the hypodissipative Navier– Stokes equations in two dimensions. Communications in Mathematical Physics , 402(1):429–446, 2023

work page 2023

[5] [4]

Bahouri, J.-Y

H. Bahouri, J.-Y. Chemin, and R. Danchin. Fourier Analysis and Nonlinear Partial Diﬀerential Equations. Springer Berlin Heidelberg, 2011

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