Pith. sign in

REVIEW 2 major objections 4 minor 108 references

Absence of blow-up in the 3D Navier-Stokes equations with transport noise

T0 review · 2 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper aims to prove that the 3D Navier–Stokes equations, driven by a suitably chosen transport noise, admit global-in-time smooth solutions with probability arbitrarily close to one, even for arbitrarily large subcritical initial data.

desk verdict First credible proof that physically motivated transport noise prevents blow-up for the true 3D Navier-Stokes equations — a major result with a transparent proof architecture and an honest accounting of its six central obstructions. read the letter →

arxiv 2607.15140 v2 pith:F3QM2LXZ submitted 2026-07-16 math.PR math-phmath.APmath.MP

Antonio Agresti This is my paper · ORCID
classification math.PRmath-phmath.APmath.MP MSC 60H5060H1576M3535Q35
keywords 3DNavier-StokesequationsregularizationbynoiseglobalsmoothnesstransportstochasticmaximalregularityCaccioppoliinequalityscalinglimitshomogenization
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 tries to establish a regularizing effect of transport noise on the 3D Navier–Stokes equations: for any prescribed large-but-finite subcritical initial datum and any small failure probability, one can choose a high-frequency, energy-preserving transport noise such that the unique maximal smooth solution is global in time with high probability. This matters because the deterministic problem is the open Millennium regularity problem, and this would be the first regularization-by-noise result for a physically motivated perturbation that does not change the energy balance. The proof shows that the mixing property of transport noise produces an effective enhanced dissipation at large scales, and it quantifies this via an exponential decay of the velocity deviation from its spatial average. A sympathetic reader would take the central claim as: transport noise, made precise through a scaling limit and homogenization, prevents finite-time blow-up of strong solutions with high probability.

What carries the argument

The central mechanism is the scaling limit: as the noise coefficients θ^N are concentrated at frequencies between N and 2N, the oscillating stochastic Stokes system homogenizes to a deterministic Stokes system with enhanced viscosity (1+3µ/5)∆. The almost-self-similarity at the microscopic scale N^{-1} gives uniform Hölder smoothness of the rescaled coefficients, which feeds the localized stochastic maximal L^p-regularity estimates. The pivotal estimate is the stochastic Caccioppoli inequality (Theorem 5.2), a reverse Poincaré inequality bounding L²-oscillations and gradients of local solutions with constants independent of N and δ; this independence relies on the Itô–Stratonovich corrector

What would settle it

Take the oscillating coefficients ζ_n = θ^N_n σ^δ_n with δN = L fixed, choose a simple divergence-free local solution such as a harmonic pressure with a constant velocity gradient, and compute the optimal constant in Theorem 5.2 as L→∞. If the constant grows with L—or if identity (5.21) fails by even a small power of N when the system is written in Itô form—the uniform micro-scale energy control collapses and Theorem 4.1 would be false.

Watch

Extended reading notes

Core claim

Theorem 4.1 asserts that for any subcritical p,q with 2/p+3/q<2, any M≥1, and any ε>0, there exist a noise intensity µ and finitely supported coefficients θ such that every divergence-free initial datum in B^{1-2/p}_{q,p} with norm at most M yields a unique maximal (p,q)-solution that is global in time with probability >1−ε, and is smooth on (0,∞)×T³. The companion Corollary 4.3 claims that transport noise induces enhanced dissipation: the L² deviation of the velocity from its mean decays exponentially with any prescribed rate λ, with finite b-th moments of the random constant on a set of probability >1−ε. The proof's weight rests on uniform-in-N estimates for an oscillating stochastic Stoke

Load-bearing premise

The stochastic Caccioppoli inequality holds with a constant independent of the oscillation parameters N and δ on every cylinder in the Avellaneda–Lin iteration; specifically, the Itô–Stratonovich correction exactly cancels the high-order pressure–velocity cross term, and if that cancellation carried any N-dependence the iteration could not reach the microscopic scale N^{-1}.

Editorial extensions

If this is right

  • If correct, arbitrarily large subcritical initial data—for example in B^{ε}_{3,8} for any ε>0—admit global smooth solutions with probability arbitrarily close to 1, so transport noise resolves the blow-up question in the stochastic setting.
  • The enhanced dissipation rate λ is prescribable in advance, and the noise intensity µ can be chosen independently of the failure probability ε; only the coefficients θ depend on ε.
  • The estimates hold uniformly on arbitrarily large balls of subcritical spaces, giving a quantitative, parameter-uniform statement rather than only an existence result.
  • The methods extend to dimensions d>3 under modified conditions, indicating the mechanism is not specific to three dimensions.
  • The scaling-limit/homogenization structure yields exponential decay of the velocity fluctuation in subcritical norms with space-time Sobolev index above −1, the threshold needed for the nonlinear cutoff argument.

Reading between the lines

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

  • A consequence the author leaves implicit: the same iteration could in principle yield quantitative large-scale regularity for other stochastic fluid systems with divergence-free transport noise, such as Boussinesq or MHD models, where energy methods also stall.
  • The proof suggests a testable quantitative prediction: for a fixed initial datum, the probability of global regularity should increase monotonically to 1 as the noise frequency N and intensity µ grow; this could be probed numerically in shell models or reduced stochastic analogues.
  • The prescribable exponential decay rate indicates an effective eddy viscosity that depends on the noise parameters; in principle one could match the rate to enhanced-mixing measurements and infer the required noise intensity.
  • A further extension, not pursued in the paper, is whether the local pressure and extrapolated-space machinery adapts to bounded domains with Dirichlet boundary conditions, which would bring the result closer to physical pipe or channel flows.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The manuscript claims a proof that the 3D Navier–Stokes equations driven by a suitably chosen transport noise admit global-in-time smooth solutions with high probability, uniformly for initial data in arbitrarily large subcritical Besov balls. The proof passes through an Itô reformulation, stochastic Caccioppoli inequalities, localized stochastic maximal regularity, quantitative homogenization/scaling limits with an Avellaneda–Lin iteration, and supremum moment bounds, culminating in a scaling-limit argument for a cut-off Navier–Stokes system. The effective viscosity coefficient 1+3µ/5 and the constants D_d are derived from the noise structure rather than fitted. The paper explicitly identifies two obstructions—loss of integrability in stochastic compactness and pressure non-locality/time-irregularity—and aims its spaces at subcritical regularity indices to overcome them.

Significance. If correct, this is a major advance in regularization by noise for the true 3D Navier–Stokes equations: it would show that a physically motivated transport noise prevents finite-time blow-up for arbitrarily large subcritical data, with an explicit enhanced-dissipation mechanism and no fitted parameters. The manuscript is unusually honest about its technical obstructions, and the proof architecture is coherent and modular. The main risks are the uniformity of the stochastic Caccioppoli inequality in the oscillation parameters and the coverage of the full subcritical range stated in Theorem 4.1 by the L^pL^q estimates proved in Section 9.

major comments (2)
  1. [§5, Theorem 5.2 and its application to (2.28)] The uniformity of the constant in (2.32) with respect to N and δ is load-bearing: the Avellaneda–Lin iteration can only reach the microscopic scale N^{-1} if the Caccioppoli constants are independent of the oscillation parameters. The algebraic cancellation in (5.21) is displayed, and the structural assumptions (5.2)–(5.4) are stated abstractly. However, the passage to the real-split rescaled coefficients of the oscillating Stokes system (2.28) is only asserted in one sentence ('Clearly...'). Please provide the explicit verification, including the values of M and ν0 for the real-split coefficients and a check that the pressure estimates in Lemmas 5.5 and 5.8 do not introduce any δ^{-1} or ||θ^N||_{ℓ∞} factor. Without this written verification, the uniformity claim is not fully documented.
  2. [§9 vs Theorem 4.1] The uniform L^pL^q estimates in Theorems 9.2 and 9.3 are proved under the restriction Sob := 1 - 2/p - 3/q ∈ [-d/2,0), see (9.3). Theorem 4.1, however, is stated for every p,q ∈ (2,∞) with 2/p + 3/q < 2, which includes cases with Sob ≥ 0. The proof of Theorem 4.1 in Section 11 is not shown in sufficient detail in the reviewed text to see how the estimates of Section 9 cover the Sob ≥ 0 range. Please either extend the estimates to that range or explain explicitly how the proof reduces the general case to the Sob < 0 setting; otherwise the statement of Theorem 4.1 exceeds the bounds proved in Section 9.
minor comments (4)
  1. [§2.1, (2.8)] The notation 'W c ãÑ X' should be defined explicitly as a compact embedding; the symbol is not standard and the reader must infer it from context.
  2. [Definition 3.1(c)] The stochastic integrand is written as '1r0,τns P[...]'; this should be '1_{[0,τ_n]}(s) P[...]' to make the indicator function unambiguous.
  3. [§5.1, (5.7)] The definition of E#_Q is understandable but could be clarified: it is a maximum of five separate expectations, not a single expectation of a maximum. Please state this explicitly to avoid confusion.
  4. [§6.1] In the blow-up proof of Corollary 6.2, the rescaled Brownian motions Ξ^{k,α}_t = N(W^{k,α}_{t0+N^{-2}(t-t0)} - W^{k,α}_{t0-N^{-2}/4}) are asserted to be standard. Please include the short covariance computation, as this rescaling is used repeatedly in later blow-up arguments.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is derived from stated hypotheses via internal estimates; self-citations are background, not load-bearing reductions.

full rationale

The derivation chain from Theorem 4.1 to the stochastic Caccioppoli inequality, localized maximal regularity, quantitative homogenization, and microscopic energy control is internal and constructive. The enhanced-viscosity coefficient 1+3µ/5 and D_d are computed from the noise via the covariance identity (3.8) and the homogenization lemmas (Lemma 7.12, Proposition 7.11), not fitted to force the blow-up conclusion. The noise parameters µ and θ are existential but are produced by explicit families θ^N with large frequency N and large intensity µ; no parameter is calibrated to data and then renamed a prediction. The self-citations, e.g., to [8, Theorem 3.2] for stochastic maximal L^p regularity or [8, Section 2] for existence of (p,q)-solutions, are prior technical inputs that do not contain the main theorem and are not used to forbid alternatives or to define the target quantity. The load-bearing uniformity issue highlighted by the skeptic — whether the constant in Theorem 5.2 is genuinely independent of N and δ — is a potential correctness risk, not a circularity: a possible failure of an estimate would collapse the proof, but the paper does not assume its conclusion as an input. No equation was found that is equivalent to its own input by construction, and there are no fitted empirical benchmarks. Therefore the verdict is no significant circularity.

Assumptions & free parameters 4 free parameters · 8 assumptions · 0 invented entities

The ledger shows a proof-first paper: the central claim is a theorem derived from stated hypotheses, not a fit to data. The honest cost concentrates in (i) the existential tuning knobs µ and N — openly disclosed in Theorem 4.1 and Remark 4.2 — and (ii) the import of the (p,q)-local theory and maximal L^p-regularity machinery from the author's own prior work [8]. No new physical entities are postulated; the transport-noise model with shell coefficients is the standard one of the Flandoli–Galeati–Luo program. The 3µ/5 effective viscosity is computed, not fitted.

free parameters (4)
  • µ (noise intensity) = µ ≥ µ₀(p,q,M), independent of ε (Remark 4.2)
    Chosen large so the homogenized viscosity 1+3µ/5 makes the deterministic limit (2.7) globally well-posed for the cutoff-removal step; an existential tuning knob the theorem is allowed to select.
  • θ (noise coefficients: frequency shell N ≤ |k| ≤ 2N, ℓ²-normalized, exponent a>0) = N ≥ N₀(p,q,M,ε,a); #{k: θ_k ≠ 0} < ∞
    The frequency scale N is the second existential knob; it controls the cumulative frequency L=δN ≥ L₀ in the homogenization and the probability margin P(τ=∞) > 1−ε. Remark 4.2 discloses that ε enters only through θ.
  • Iteration geometry (η, L₀) and Hölder gain γ₀ = η ∈ (0,1/4], γ₀ ∈ (0,(1−2/p−d/q)∧(1−2/p₀)), L₀ ≥ 1
    Hand-chosen constants in the one-step improvement (Lemma 8.3) and blow-up iteration (Lemma 8.6); they set the contraction rate η^{ℓγ₀} and therefore the reach of the energy control down to the microscopic scale.
  • Exponents (p,q) and proof parameters (p₀,q₀,r) = p,q ∈ (2,∞), 2/p+3/q<2; p₀>2p, q₀>2q, r≫1
    User-selected regularity class for the data (4.1)–(4.2). The proof parameters p₀, q₀, r compensate the suboptimality of the supremum-moment bound (2.20) ('need for high moments r ≫ 1').
assumptions (8)
  • domain assumption Local (p,q)-solution theory up to a positive lifetime for (1.1) (Definition 3.1), for subcritical data
    Existence, uniqueness and regularity up to a positive lifetime are imported from [8, §2] (the author's own prior work); the present paper supplies only the global-in-time statement.
  • ad hoc to paper Noise structure: σ_{k,α} = a_{k,α}e^{2πik·x}, complex Brownian motions with W^{k,α} = conj(W^{−k,α}), radially symmetric normalized θ (3.1)
    The specific shell-concentrated coefficient family is chosen to create the almost-self-similarity at scale N^{-1}; it is inherited from the scaling-limit program [48,49,55] and the physical separation-of-scales motivation in §1.2 is heuristic.
  • standard math Radial-symmetry/2/3 identity (3.8): Σ_{k,α} θ²ₖ σ_{k,α}⊗σ_{−k,α} = (2/3)Id for normalized radially symmetric θ
    Algebraic identity for the Fourier basis; converts the Stratonovich noise into Itô form with diffusion µΔ and drives the 3µ/5 effective-viscosity coefficient.
  • standard math Bessel-type inequality (Lemma 7.12): rate (δN)^{−d} for the rescaled noise coefficients
    Parseval computation for the incomplete system {δ^{d/2}e^{−2πiδx·k}a_{k,α}} on the δ^{-1}-box; the source of the cumulative-frequency decay used throughout Proposition 7.11.
  • standard math Extrapolated Helmholtz projection P_{−1,B}: X^Dir_{−1}(B;R^d) → X^Sto_{−1}(B) is bounded (Lemma 7.7)
    Proved in the paper following [75, Prop. 9.14]; essential for writing the difference SPDE in extrapolated spaces and for the local mixing estimates (§7.2–7.3).
  • domain assumption Deterministic 3D NSEs with enhanced viscosity 1+3µ/5 are globally well-posed for µ ≫ 1, with bounds in X = L^p_t L^q_x of Sobolev index ≥ −1
    Classical large-viscosity/small-data theory (Fujita–Kato type); stated without proof or citation in §2.1 and used to remove the cutoff by choosing R ≫ 1.
  • standard math Stochastic maximal L^p-regularity (global version [8, Thm 3.2]; Krylov; van Neerven–Veraar–Weis)
    Background SPDE tool used to localize (6.1)→(6.5) and throughout §7; the paper notes the extension to r-th moments follows from [98, §3.5] and [96, Cor. 7.4].
  • domain assumption Subcriticality conditions: 2/p+3/q<2 for the data class (4.1) and 2/p+d/q<1 for the pointwise/supremum bounds (2.17)
    These regimes define the theorem's scope; the paper notes (2.17) is necessary even in the absence of noise, so the constraint is honest rather than ad hoc.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Absence of blow-up in the 3D Navier-Stokes equations with transport noise." pith.science (2026). https://pith.science/paper/F3QM2LXZ

@misc{pith2026260715140,
  author       = {Pith},
  title        = {Pith review of: Absence of blow-up in the 3D Navier-Stokes equations with transport noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F3QM2LXZ}},
  note         = {Machine review of arXiv:2607.15140}
}
read the original abstract

Establishing the global-in-time smoothness of solutions to the 3D Navier-Stokes equations (NSEs) with large initial data remains a long-standing open problem. In this paper, we prove that the 3D NSEs driven by a suitably chosen transport noise -- a physically motivated stochastic perturbation -- admit global-in-time smooth solutions with high probability. This regularizing effect holds uniformly for initial data in arbitrarily large balls of subcritical function spaces with positive smoothness.

Figures

Figures reproduced from arXiv: 2607.15140 by the authors.

Figure 1
Figure 1. Proof architecture. The main result (Theorem 4.1) and the crucial intermediate step (Theorem 9.4) are highlighted in bold. Relevant subsections are indicated in parentheses. 2. Proof outline This section provides an overview of the proof strategy for Theorem 4.1, which generalizes Theorem 1.1. The architecture is summarized in [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Uniform smoothness at the microscopic scale, where N1 " N0. Localized stochastic maximal L p -regularity and supremum moment bounds. Let Q be a parabolic cylinder with center pt0, x0q. The just-discussed almost self-similarity implies that the coefficients of the stochastic Stokes system (2.11) at the microscopic scale N ´1Q are uniformly-in-N bounded, for example, in H¨older continuous norms, see (2.15). In the lat… view at source ↗
Figure 3
Figure 3. Construction in the proof of the estimate (9.15) where I “ pt0 ´ 1{4, t0q. Applying Corollary 6.2 to the local solution ´ vp N , πp N ´ [PITH_FULL_IMAGE:figures/full_fig_p095_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

108 extracted references · 3 linked inside Pith

  1. [8]

    Agresti and M.C

    A. Agresti and M.C. Veraar. Stochastic Navier-Stokes equations for turbulent flows in critical spaces.Comm. Math. Phys., 405(2):Paper No. 43, 57, 2024

    2024

  2. [1]

    A. Agresti. Delayed blow-up and enhanced diffusion by transport noise for systems of reaction- diffusion equations.Stoch. Partial Differ. Equ. Anal. Comput., 12(3):1907–1981, 2024

    1907

  3. [2]

    A. Agresti. Global smooth solutions by transport noise of 3D Navier-Stokes equations with small hyperviscosity.arXiv preprint arXiv:2406.09267, 2024. To appear in Ann. Probab

    arXiv 2024

  4. [3]

    A. Agresti. On anomalous dissipation induced by transport noise.Math. Ann., 393(3-4):3141– 3190, 2025

    2025

  5. [4]

    A. Agresti. On the role ofL ppLqq-techniques in regularization by noise of reaction-diffusion equations. InStochastic geophysical fluid dynamics, volume 55 ofOberwolfach Semin., pages 59–87. Birkh¨ auser-Springer, 2025

    2025

  6. [5]

    Agresti, F

    A. Agresti, F. Butori, and E. Luongo. Global smooth solutions by high mode Lie- transport noise for logarithmically hyperdissipative Navier-Stokes equations.arXiv preprint arXiv:2606.28575, 2026. ABSENCE OF BLOW-UP IN 3D NSEs WITH TRANSPORT NOISE 125

    arXiv 2026

  7. [6]

    Agresti, M

    A. Agresti, M. Kniely, and B.Q. Tang. Global classical solutions by transport noise of reaction- diffusion systems with entropy dissipation. In preparation, 2026

    2026

  8. [7]

    Agresti and M

    A. Agresti and M. Sauerbrey. Well-posedness of the stochastic thin-film equation with an interface potential.Comm. Math. Phys., 407(8):158, 2026

    2026

Show all 108 references
  1. [9]

    Agresti and M.C

    A. Agresti and M.C. Veraar. Nonlinear SPDEs and Maximal Regularity: An Extended Survey. NoDEA Nonlinear Differential Equations Appl., 32(6):Paper No. 123, 2025

    2025

  2. [10]

    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. [11]

    Albritton and M

    D. Albritton and M. Colombo. Non-uniqueness of Leray solutions to the hypodissipa- tive Navier–Stokes equations in two dimensions.Communications in Mathematical Physics, 402(1):429–446, 2023

    2023

  4. [12]

    Amann.Linear and quasilinear parabolic problems

    H. Amann.Linear and quasilinear parabolic problems. Vol. I, Abstract linear theory, volume 89 ofMonographs in Mathematics. Birkh¨ auser Boston Inc., Boston, MA, 1995

    1995

  5. [13]

    Armstrong, T

    S. Armstrong, T. Kuusi, and J.-C. Mourrat.Quantitative stochastic homogenization and large- scale regularity, volume 352 ofGrundlehren der mathematischen Wissenschaften. Springer, Cham, 2019

    2019

  6. [14]

    Armstrong and V

    S. Armstrong and V. Vicol. Anomalous diffusion by fractal homogenization.Ann. PDE, 11(1):Paper No. 2, 145, 2025

    2025

  7. [15]

    L. Arnold. Stabilization by noise revisited.Z. Angew. Math. Mech., 70(7):235–246, 1990

    1990

  8. [16]

    Arnold, H

    L. Arnold, H. Crauel, and V. Wihstutz. Stabilization of linear systems by noise.SIAM J. Control Optim., 21(3):451–461, 1983

    1983

  9. [17]

    Avellaneda and F.-H

    M. Avellaneda and F.-H. Lin. Compactness methods in the theory of homogenization.Comm. Pure Appl. Math., 40(6):803–847, 1987

    1987

  10. [18]

    Bagnara, L

    M. Bagnara, L. Galeati, and M. Maurelli. Regularization by rough Kraichnan noise for the generalised SQG equations.Math. Ann., 392(4):4773–4830, 2025

    2025

  11. [19]

    Bechtold and M

    F. Bechtold and M. Hofmanov´ a. Can probability theory really help tame problems in math- ematical hydrodynamics?Mitteilungen der Deutschen Mathematiker-Vereinigung, 30(4):231– 234, 2022

    2022

  12. [20]

    Bergh and J

    J. Bergh and J. L¨ ofstr¨ om.Interpolation spaces. An introduction. Springer-Verlag, Berlin, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223

    1976

  13. [21]

    Boussinesq.Essai sur la th´ eorie des eaux courantes, volume 23 ofM´ emoires pr´ esent´ es par divers savants ` a l’Acad´ emie des Sciences

    J. Boussinesq.Essai sur la th´ eorie des eaux courantes, volume 23 ofM´ emoires pr´ esent´ es par divers savants ` a l’Acad´ emie des Sciences. Imprimerie Nationale, Paris, 1877

  14. [22]

    Brezis.Functional analysis, Sobolev spaces and partial differential equations

    H. Brezis.Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011

    2011

  15. [23]

    Bru` e, M

    E. Bru` e, M. Colombo, and A. Kumar. Sharp nonuniqueness in the transport equation with sobolev velocity field.J. Eur. Math. Soc., 2026. Published online first

    2026

  16. [24]

    Brze´ zniak, M

    Z. Brze´ zniak, M. Capi´ nski, and F. Flandoli. Stochastic partial differential equations and tur- bulence.Math. Models Methods Appl. Sci., 1(1):41–59, 1991

    1991

  17. [25]

    Buckmaster and V

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

    2019

  18. [26]

    Butori, F

    F. Butori, F. Flandoli, E. Luongo, and Y. Tahraoui. Background Vlasov equations and Young measures for passive scalar and vector advection equations under special stochastic scaling limits.Probability Theory and Related Fields, pages 1–62, 2026

    2026

  19. [27]

    Caffarelli, R

    L. Caffarelli, R. Kohn, and L. Nirenberg. Partial regularity of suitable weak solutions of the Navier-Stokes equations.Comm. Pure Appl. Math., 35(6):771–831, 1982

    1982

  20. [28]

    Coghi and M

    M. Coghi and M. Maurelli. Existence and uniqueness by Kraichnan noise for 2D Euler equations with unbounded vorticity.Nonlinearity, 39(5):Paper No. 055003, 52, 2026

    2026

  21. [29]

    Colombo, C

    M. Colombo, C. De Lellis, and A. Massaccesi. The generalized Caffarelli-Kohn-Nirenberg the- orem for the hyperdissipative Navier-Stokes system.Communications on Pure and Applied Mathematics, 73(3):609–663, 2020

    2020

  22. [30]

    Colombo and S

    M. Colombo and S. Haffter. Global regularity for the hyperdissipative Navier-Stokes equation below the critical order.J. Differential Equations, 275:815–836, 2021. 126 ANTONIO AGRESTI

    2021

  23. [31]

    Constantin, A

    P. Constantin, A. Kiselev, L. Ryzhik, and A. Zlatoˇ s. Diffusion and mixing in fluid flow.Ann. of Math. (2), 168(2):643–674, 2008

    2008

  24. [32]

    Coti Zelati, G

    M. Coti Zelati, G. Crippa, G. Iyer, and A.L. Mazzucato. Mixing in incompressible flows: transport, dissipation, and their interplay.Notices Amer. Math. Soc., 71(5):593–604, 2024

    2024

  25. [33]

    Da Prato and A

    G. Da Prato and A. Debussche. Ergodicity for the 3D stochastic Navier-Stokes equations.J. Math. Pures Appl. (9), 82(8):877–947, 2003

    2003

  26. [34]

    Da Prato and J

    G. Da Prato and J. Zabczyk.Stochastic equations in infinite dimensions, volume 152 ofEncy- clopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, second edition, 2014

    2014

  27. [35]

    De Lellis and L

    C. De Lellis and L. Sz´ ekelyhidi, Jr. The Euler equations as a differential inclusion.Ann. of Math. (2), 170(3):1417–1436, 2009

    2009

  28. [36]

    Debussche and E

    A. Debussche and E. M´ emin. Variational principles for fully coupled stochastic fluid dynamics across scales.Phys. D, 481:Paper No. 134777, 11, 2025

    2025

  29. [37]

    Debussche and U

    A. Debussche and U. Pappalettera. Second order perturbation theory of two-scale systems in fluid dynamics.J. Eur. Math. Soc., 28(4):1533–1595, 2026

    2026

  30. [38]

    Denk and M

    R. Denk and M. Kaip.General parabolic mixed order systems inL p and applications, volume 239 ofOperator Theory: Advances and Applications. Birkh¨ auser/Springer, Cham, 2013

    2013

  31. [39]

    R. J. DiPerna and P.-L. Lions. Ordinary differential equations, transport theory and Sobolev spaces.Invent. Math., 98(3):511–547, 1989

    1989

  32. [40]

    Evans and R.F

    L.C. Evans and R.F. Gariepy.Measure theory and fine properties of functions. Textbooks in Mathematics. CRC Press, Boca Raton, FL, revised edition, 2015

    2015

  33. [41]

    Fefferman

    C.L. Fefferman. Existence and smoothness of the Navier-Stokes equation.The millennium prize problems, 57:67, 2000

    2000

  34. [42]

    Flandoli.Random perturbation of PDEs and fluid dynamic models, volume 2015 ofLecture Notes in Mathematics

    F. Flandoli.Random perturbation of PDEs and fluid dynamic models, volume 2015 ofLecture Notes in Mathematics. Springer, Heidelberg, 2011. Lectures from the 40th Probability Summer School held in Saint-Flour, 2010, ´Ecole d’ ´Et´ e de Probabilit´ es de Saint-Flour

    2015

  35. [43]

    Flandoli

    F. Flandoli. A stochastic view over the open problem of well-posedness for the 3D Navier- Stokes equations. InStochastic analysis: a series of lectures, volume 68 ofProgr. Probab., pages 221–246. Birkh¨ auser/Springer, Basel, 2015

    2015

  36. [44]

    Flandoli

    F. Flandoli. Stochastic fluid mechanics and Boussinesq hypothesis. InStochastic geophysical fluid dynamics, volume 55 ofOberwolfach Semin., pages 1–13. Birkh¨ auser/Springer, Cham, 2025

    2025

  37. [45]

    Flandoli, L

    F. Flandoli, L. Galeati, and D. Luo. Delayed blow-up by transport noise.Comm. Partial Differential Equations, 46(9):1757–1788, 2021

    2021

  38. [46]

    Flandoli, L

    F. Flandoli, L. Galeati, and D. Luo. Eddy heat exchange at the boundary under white noise turbulence.Philosophical Transactions of the Royal Society A, 380(2219):20210096, 2022

    2022

  39. [47]

    Flandoli, L

    F. Flandoli, L. Galeati, and D. Luo. Quantitative convergence rates for scaling limit of SPDEs with transport noise.Journal of Differential Equations, 394:237–277, 2024

    2024

  40. [48]

    Flandoli, M

    F. Flandoli, M. Gubinelli, and E. Priola. Well-posedness of the transport equation by stochastic perturbation.Invent. Math., 180(1):1–53, 2010

    2010

  41. [49]

    Flandoli and D

    F. Flandoli and D. Luo. High mode transport noise improves vorticity blow-up control in 3D Navier–Stokes equations.Probab. Theory Related Fields, 180(1–2):309–363, 2021

    2021

  42. [50]

    Flandoli and D

    F. Flandoli and D. Luo. On the Boussinesq hypothesis for a stochastic Proudman-Taylor model.SIAM J. Math. Anal., 56(3):3886–3923, 2024

    2024

  43. [51]

    Flandoli and E

    F. Flandoli and E. Luongo.Stochastic partial differential equations in fluid mechanics, volume

  44. [52]

    Flandoli and M

    F. Flandoli and M. Romito. Partial regularity for the stochastic Navier-Stokes equations. Trans. Amer. Math. Soc., 354(6):2207–2241, 2002

    2002

  45. [53]

    Flandoli and M

    F. Flandoli and M. Romito. Markov selections for the 3D stochastic Navier-Stokes equations. Probab. Theory Related Fields, 140(3-4):407–458, 2008

    2008

  46. [54]

    Galdi.An introduction to the mathematical theory of the Navier-Stokes equations

    G.P. Galdi.An introduction to the mathematical theory of the Navier-Stokes equations. Springer Monographs in Mathematics. Springer, New York, second edition, 2011. Steady-state problems. ABSENCE OF BLOW-UP IN 3D NSEs WITH TRANSPORT NOISE 127

    2011

  47. [55]

    L. Galeati. On the convergence of stochastic transport equations to a deterministic parabolic one.Stochastics and Partial Differential Equations: Analysis and Computations, 8(4):833– 868, 2020

    2020

  48. [56]

    Geiss and M

    S. Geiss and M. Scheutzow. Sharpness of Lenglart’s domination inequality and a sharp mono- tone version.Electron. Commun. Probab., 26:Paper No. 44, 8, 2021

    2021

  49. [57]

    Giaquinta.Multiple integrals in the calculus of variations and nonlinear elliptic systems, volume 105 ofAnnals of Mathematics Studies

    M. Giaquinta.Multiple integrals in the calculus of variations and nonlinear elliptic systems, volume 105 ofAnnals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1983

    1983

  50. [58]

    Giaquinta and L

    M. Giaquinta and L. Martinazzi.An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, volume 11 ofLecture Notes. Scuola Normale Superiore di Pisa (New Series). Edizioni della Normale, Pisa, second edition, 2012

    2012

  51. [59]

    Glatt-Holtz and V

    N.E. Glatt-Holtz and V. Vicol. Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise.Ann. Probab., 42(1):80–145, 2014

    2014

  52. [60]

    Grafakos.Classical Fourier analysis, volume 249 ofGraduate Texts in Mathematics

    L. Grafakos.Classical Fourier analysis, volume 249 ofGraduate Texts in Mathematics. Springer, New York, third edition, 2014

    2014

  53. [61]

    Haase.The functional calculus for sectorial operators and similarity methods

    M.H.A. Haase.The functional calculus for sectorial operators and similarity methods. PhD thesis, Universit¨ at Ulm, 2003

    2003

  54. [62]

    Hofmanov´ a, T

    M. Hofmanov´ a, T. Lange, and U. Pappalettera. Global existence and non-uniqueness of 3D Eu- ler equations perturbed by transport noise.Probability Theory and Related Fields, 188(3):1183– 1255, 2024

    2024

  55. [63]

    Hofmanov´ a, J.-M

    M. Hofmanov´ a, J.-M. Leahy, and T. Nilssen. On a rough perturbation of the Navier-Stokes system and its vorticity formulation.Ann. Appl. Probab., 31(2):736–777, 2021

    2021

  56. [64]

    Hofmanov´ a, U

    M. Hofmanov´ a, U. Pappalettera, R. Zhu, and X. Zhu. Anomalous and total dissipation due to advection by solutions of randomly forced Navier-Stokes equations.Ann. Appl. Probab., 35(5):3119–3149, 2025

    2025

  57. [65]

    Hofmanov´ a, R

    M. Hofmanov´ a, R. Zhu, and X. Zhu. Nonuniqueness in law of stochastic 3D Navier-Stokes equations.J. Eur. Math. Soc. (JEMS), 26(1):163–260, 2024

    2024

  58. [66]

    D.D. Holm. Variational principles for stochastic fluid dynamics.Proceedings of the Royal So- ciety A: Mathematical, Physical and Engineering Sciences, 471(2176):20140963, 2015

    2015

  59. [67]

    T. Hou, Y. Wang, and C. Yang. Nonuniqueness of Leray-Hopf solutions to the unforced in- compressible 3D Navier-Stokes equation.arXiv preprint arXiv:2509.25116, 2025

    2025

  60. [68]

    Hyt¨ onen, J.M.A.M

    T. Hyt¨ onen, J.M.A.M. van Neerven, M.C. Veraar, and L. Weis.Analysis in Banach spaces. Vol. III. Harmonic analysis and spectral theory, volume 76 ofErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge.Springer, 2023

    2023

  61. [69]

    Hyt¨ onen, J.M.A.M

    T.P. Hyt¨ onen, J.M.A.M. van Neerven, M.C. Veraar, and L. Weis.Analysis in Banach spaces. Vol. II. Probabilistic Methods and Operator Theory., volume 67 ofErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge.Springer, 2017

    2017

  62. [70]

    Hyt¨ onen, J.M.A.M

    T.P. Hyt¨ onen, J.M.A.M. van Neerven, M.C. Veraar, and L.W. Weis.Analysis in Banach spaces. Vol. I. Martingales and Littlewood-Paley theory, volume 63 ofErgebnisse der Mathe- matik und ihrer Grenzgebiete. 3. Folge.Springer, 2016

    2016

  63. [71]

    P. Isett. A proof of Onsager’s conjecture.Ann. of Math. (2), 188(3):871–963, 2018

    2018

  64. [72]

    G. Iyer, X. Xu, and A. Zlatoˇ s. Convection-induced singularity suppression in the Keller-Segel and other non-linear PDEs.Trans. Amer. Math. Soc., 374(9):6039–6058, 2021

    2021

  65. [73]

    Jia and V

    H. Jia and V. Sverak. Are the incompressible 3d Navier-Stokes equations locally ill-posed in the natural energy space?J. Funct. Anal., 268(12):3734–3766, 2015

    2015

  66. [74]

    Jiao and D

    S. Jiao and D. Luo. Well-posedness of stochastic mSQG equations with Kraichnan noise and Lp data.J. Differential Equations, 438:Paper No. 113362, 41, 2025

    2025

  67. [75]

    Kalton, P.C

    N.J. Kalton, P.C. Kunstmann, and L.W. Weis. Perturbation and interpolation theorems for theH 8-calculus with applications to differential operators.Math. Ann., 336(4):747–801, 2006

    2006

  68. [76]

    Kiselev and X

    A. Kiselev and X. Xu. Suppression of chemotactic explosion by mixing.Arch. Ration. Mech. Anal., 222(2):1077–1112, 2016

    2016

  69. [77]

    N.V. Krylov. An analytic approach to SPDEs. InStochastic partial differential equations: six perspectives, volume 64 ofMath. Surveys Monogr., pages 185–242. Amer. Math. Soc., Providence, RI, 1999. 128 ANTONIO AGRESTI

    1999

  70. [78]

    Krylov.Lectures on elliptic and parabolic equations in Sobolev spaces, volume 96 of Graduate Studies in Mathematics

    N.V. Krylov.Lectures on elliptic and parabolic equations in Sobolev spaces, volume 96 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2008

    2008

  71. [79]

    T. Lange. Regularization by noise of an averaged version of the Navier–Stokes equations.J. Dyn. Differ. Equ., 36(4):3011–3036, 2024

    2024

  72. [80]

    P. G. Lemari´ e-Rieusset.The Navier-Stokes problem in the 21st century. CRC Press, Boca Raton, FL, 2016

    2016

  73. [81]

    Lenglart

    E. Lenglart. Relation de domination entre deux processus.Ann. Inst. H. Poincar´ e Sect. B (N.S.), 13(2):171–179, 1977

    1977

  74. [82]

    Lions.Quelques m´ ethodes de r´ esolution des probl` emes aux limites non lin´ eaires

    J.-L. Lions.Quelques m´ ethodes de r´ esolution des probl` emes aux limites non lin´ eaires. Dunod, 1969

    1969

  75. [83]

    Lions and E

    J.-L. Lions and E. Magenes.Non-homogeneous boundary value problems and applications. Vol. I. Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth

    1972

  76. [84]

    Lions and E

    J.-L. Lions and E. Magenes.Non-homogeneous boundary value problems and applications. Vol. II. Die Grundlehren der mathematischen Wissenschaften, Band 182. Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth

    1972

  77. [85]

    Liu and M

    W. Liu and M. R¨ ockner.Stochastic partial differential equations: an introduction. Universi- text. Springer, Cham, 2015

    2015

  78. [86]

    L¨ u and J.M.A.M

    Q. L¨ u and J.M.A.M. van Neerven. On conditional expectations inLppµ;L qpν;Xqq.Positivity, 23(1):11–19, 2019

    2019

  79. [87]

    Lunardi.Interpolation theory

    A. Lunardi.Interpolation theory. Appunti. Scuola Normale Superiore di Pisa. Edizioni della Normale, Pisa, second edition, 2009

    2009

  80. [88]

    D. Luo. Enhanced Dissipation for Stochastic Navier–Stokes Equations with Transport Noise. J. Dynam. Differential Equations, 37(1):859–894, 2025

    2025

  81. [89]

    D. Luo, B. Tang, and G. Zhao. An elementary approach to mixing and dissipation enhancement by transport noise.arXiv preprint arXiv:2402.07484, 2024. To appear in Ann. Inst. Henri Poincar´ e Probab. Stat

    2024 arXiv

  82. [90]

    Majda, I

    A.J. Majda, I. Timofeyev, and E. Vanden Eijnden. A mathematical framework for stochastic climate models.Comm. Pure Appl. Math., 54(8):891–974, 2001

    2001

  83. [91]

    Mazzucato, Y

    A.L. Mazzucato, Y. Feng, and C. Nobili. Enhanced dissipation by advection and applications to PDEs.Appl. Math. Model., 150(part A):Paper No. 116310, 24, 2026

    2026

  84. [92]

    E. M´ emin. Fluid flow dynamics under location uncertainty.Geophys. Astrophys. Fluid Dyn., 108(2):119–146, 2014

    2014

  85. [93]

    Mikulevicius and B

    R. Mikulevicius and B. Rozovskii. On equations of stochastic fluid mechanics. InStochastics in finite and infinite dimensions, Trends Math., pages 285–302. Birkh¨ auser Boston, Boston, MA, 2001

    2001

  86. [94]

    Mikulevicius and B.L

    R. Mikulevicius and B.L. Rozovskii. Stochastic Navier–Stokes equations for turbulent flows. SIAM Journal on Mathematical Analysis, 35(5):1250–1310, 2004

    2004

  87. [95]

    van Neerven, M.C

    J.M.A.M. van Neerven, M.C. Veraar, and L.W. Weis. Stochastic maximalL p-regularity.Ann. Probab., 40(2):788–812, 2012

    2012

  88. [96]

    van Neerven, M.C

    J.M.A.M. van Neerven, M.C. Veraar, and L.W. Weis. Stochastic integration in Banach spaces— a survey. InStochastic analysis: a series of lectures, volume 68 ofProgr. Probab., pages 297–332. Birkh¨ auser/Springer, Basel, 2015

    2015

  89. [97]

    Pappalettera

    U. Pappalettera. Global existence and non-uniqueness for the Cauchy problem associated to 3D Navier-Stokes equations perturbed by transport noise.Stoch. Partial Differ. Equ. Anal. Comput., 12(3):1769–1804, 2024

    2024

  90. [98]

    Portal and M.C

    P. Portal and M.C. Veraar. Stochastic maximal regularity for rough time-dependent problems. Stoch. Partial Differ. Equ. Anal. Comput., 7(4):541–597, 2019

    2019

  91. [99]

    Pr¨ uss and G

    J. Pr¨ uss and G. Simonett.Moving interfaces and quasilinear parabolic evolution equations, volume 105 ofMonographs in Mathematics. Birkh¨ auser/Springer, 2016

    2016

  92. [100]

    Ruzhansky and V

    M. Ruzhansky and V. Turunen. Quantization of pseudo-differential operators on the torus.J. Fourier Anal. Appl., 16(6):943–982, 2010

    2010

  93. [101]

    Scheffer

    V. Scheffer. Hausdorff measure and the Navier-Stokes equations.Comm. Math. Phys., 55(2):97–112, 1977. ABSENCE OF BLOW-UP IN 3D NSEs WITH TRANSPORT NOISE 129

    1977

  94. [102]

    Schmeisser and H

    H.J. Schmeisser and H. Triebel.Topics in Fourier Analysis and Function Spaces. Wiley, 1987

    1987

  95. [103]

    Shen.Periodic homogenization of elliptic systems, volume 269 ofOperator Theory: Ad- vances and Applications

    Z. Shen.Periodic homogenization of elliptic systems, volume 269 ofOperator Theory: Ad- vances and Applications. Birkh¨ auser/Springer, Cham, 2018. Advances in Partial Differential Equations (Basel)

    2018

  96. [104]

    T. Tao. Finite time blowup for an averaged three-dimensional Navier-Stokes equation.J. Amer. Math. Soc., 29(3):601–674, 2016

    2016

  97. [105]

    M. E. Taylor.Partial differential equations I. Basic theory, volume 115 ofApplied Mathemat- ical Sciences. Springer, New York, second edition, 2011

    2011

  98. [106]

    Taylor.Partial differential equations II

    M.E. Taylor.Partial differential equations II. Qualitative studies of linear equations, volume 116 ofApplied Mathematical Sciences. Springer, New York, second edition, 2011

    2011

  99. [107]

    Veraar and L

    M. Veraar and L. Weis. A note on maximal estimates for stochastic convolutions.Czechoslovak Math. J., 61(136)(3):743–758, 2011. Department of Mathematics Guido Castelnuovo, Sapienza University of Rome, P.le Aldo Moro 5, 00185 Rome, Italy. Email address:antonio.agresti@uniroma1.it

    2011

  100. [2330]

    Springer Nature, 2023

    2023

Pith tools

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