Pith. sign in

REVIEW 2 major objections 4 minor 1 cited by

Scaling Limit and Large Deviation for 3D Globally Modified Stochastic Navier-Stokes Equations with Transport Noise

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read This paper proves that transport noise spread across many high Fourier modes drives the 3D globally modified stochastic Navier-Stokes equations to a deterministic equation with enhanced dissipation, and establishes a large deviation…

desk verdict Solid extension of the transport-noise scaling-limit and LDP machinery to 3D globally modified Navier-Stokes equations, but the LDP theorem overreaches: the proof needs Λ+r>5/2, and the quantitative rate has a typo. read the letter →

arxiv 2412.20752 v2 pith:DYWZYY33 submitted 2024-12-30 math.PR

Chang Liu , Dejun Luo This is my paper · ORCID
classification math.PR MSC 60H1535Q3060F1076D05
keywords 3DgloballymodifiedNavier-Stokesequationstransportnoisewell-posednessscalinglimitlargedeviationprincipleenhanceddissipationpathwiseuniqueness
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 studies a three-dimensional fluid model in which the Navier-Stokes nonlinearity is multiplied by a cutoff that keeps the equations globally well-posed, while the fluid is shaken by transport noise, meaning noise advected along the velocity field. It establishes three results: the stochastic system has a pathwise unique weak solution for any $L^2$ initial datum; when the noise is spread over more and more high Fourier modes, the random solutions converge strongly in $L^p(\Omega; X)$ to the solution of a deterministic Navier-Stokes-type equation with an extra dissipation term $\frac{3\nu}{5}\Delta$; and in the hyperviscous case the laws satisfy a uniform large deviation principle whose rate function is expressed through a skeleton equation. If correct, the results make precise a mechanism by which small-scale turbulent fluctuations, modeled as transport noise, act on large scales as enhanced viscosity.

What carries the argument

The load-bearing mechanism is the combination of the cutoff function $F_N(\|u\|_{H^{1-\delta}})=\min\{1,N/\|u\|_{H^{1-\delta}}\}$ with high-mode transport noise. The cutoff makes the nonlinearity globally tame, giving uniform energy bounds and, via the known uniqueness strategy for the deterministic globally modified Navier-Stokes equations, pathwise uniqueness for $L^2$ initial data. The noise coefficients $\theta_n$ concentrate on high modes in the sense $\|\theta_n\|_{\ell^2}=1$, $\|\theta_n\|_{\ell^\infty}\to0$; in the Stratonovich formulation the It\^o correction $S_{\theta_n}$ then tends to $\frac{3\nu}{5}\Delta$, quantitatively through Theorem 2.2, and this is the mechanism converting random small-scale advection into deterministic large-scale dissipation. For the large deviation principle, the skeleton equation $\partial_t u+F_N(\|u\|_{H^1})\Pi(u\cdot\nabla u)=-(-\Delta)^\Lambda u+\frac{3\nu}{5}\Delta u+\Pi(g\cdot\nabla u)$ carries the argument: the rate function is the minimal $H^r$ cost of a control $g$ that produces a given path.

What would settle it

Take $\Lambda=1.2$ and $r=0.5$, so $\Lambda+r=1.7<2.5$, and look for two weak solutions of the skeleton equation with the same $u_0\in L^2$ and $g\in L^2([0,T];H^r)$. If such a non-unique pair exists, $I_{u_0}$ is not a single-valued rate function and Theorem 1.6 fails on its stated range; if no such pair exists, one needs a proof that does not rely on the bound $\|\Pi(g\cdot\nabla\xi)\|_{H^{-\Lambda}}\lesssim\|g\|_{H^r}\|\xi\|_{H^{5/2-\Lambda-r}}$, which becomes unavailable when $\Lambda+r\le5/2$.

Watch

Extended reading notes

Core claim

The central discovery is that the Stratonovich transport noise in the globally modified 3D Navier-Stokes system is not merely a perturbation: when the noise coefficients $\theta_n$ are normalized by $\|\theta_n\|_{\ell^2}=1$ and spread over an increasing number of modes, so that $\|\theta_n\|_{\ell^\infty}\to 0$ with the explicit choice $\theta^k_n=\sqrt{\varepsilon_n}|k|^{-r}\mathbf{1}_{1\le|k|\le n}$, the Stratonovich-It\^o corrector $S_{\theta_n}$ converges to $\frac{3\nu}{5}\Delta$. Consequently the stochastic equations converge strongly in $L^p(\Omega;X)$ to the deterministic GMNSE with the extra dissipation $\frac{3\nu}{5}\Delta$, for every $\Lambda\in[1,2)$ and $T>0$. In the hyperviscous regime $\Lambda\in(1,2)$, $\delta=0$, the paper proves a uniform large deviation principle with rate function $I_{u_0}(u)=\inf_{g: u=G_0(u_0,\mathrm{Int}(g))}\frac12\int_0^T\|g(s)\|^2_{H^r}\,ds$, stated for $r\in(0,\tfrac32)$; the well-posedness of the skeleton equation that supports this rate function is proved under the stricter condition $\Lambda+r>\tfrac52$.

Load-bearing premise

The load-bearing premise is that the skeleton equation is well posed on the same parameter range where the LDP is stated: the proof of uniqueness requires $\Lambda+r>5/2$, a condition not stated in Theorem 1.6, so if $\Lambda+r\le5/2$ the rate function may not be well-defined and the LDP is not justified.

Editorial extensions

If this is right

  • Pathwise uniqueness holds for the stochastic GMNSE with transport noise from every $u_0\in L^2$, for $\Lambda\in[1,2)$; in particular, the solution is probabilistically strong.
  • Spreading the noise over higher modes makes the random solutions converge strongly in $L^p(\Omega;X)$ to the deterministic GMNSE with viscosity enhanced by $\frac{3\nu}{5}$; the noise acts as dissipation on the mean flow.
  • Initial data convergence $u_n(0)\rightharpoonup \bar u(0)$ in $L^2$ is enough for the scaling limit, so the result is not tied to identical initial data.
  • In the hyperviscous case the laws of the solutions satisfy a uniform large deviation principle; rare events are governed by the minimal control $g$ needed to produce a given path through the skeleton equation.
  • Under the additional regularity $u_0\in H^1$ and a large enough choice of $\nu$, the mean-square $H^{1-\delta}$ error has an explicit rate in $n^{-2\delta}$ and in a power of $\|\theta_n\|_{\ell^\infty}$, making the scaling limit quantitative.

Reading between the lines

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

  • The LDP is stated for every $r\in(0,3/2)$, but the skeleton-equation well-posedness in the proof requires $\Lambda+r>5/2$; unless that missing argument is supplied, the theorem should be read as established on the narrower range $\Lambda+r>5/2$.
  • The same high-mode concentration mechanism likely transfers to other globally modified or tamed fluid models, because the proof uses only the Lipschitz-type property of the cutoff and the corrector convergence $S_{\theta_n}\to\frac{3\nu}{5}\Delta$.
  • The restriction $\delta>0$ when $\Lambda=1$ is an artifact of the compactness proof for the nonlinear term; if a stronger convergence in $H^1$ could be obtained, the scaling limit would cover the original GMNSE cutoff norm $\|u\|_{H^1}$.
  • The quantitative rate in Theorem 4.2 suggests a concrete recipe for numerical closure: choose the cutoff scale $N$ and a sufficiently large $\nu$, then let the noise spread over modes $|k|\le n$; the error to the deterministic limit is then controlled by $n^{-2\delta}$.
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 paper studies the 3D globally modified stochastic Navier-Stokes equations with transport noise on the torus, allowing hyperviscosity Λ ∈ [1,2). It claims three main results: (i) existence and pathwise uniqueness of weak solutions for L² initial data (Theorem 1.2); (ii) a scaling limit in which noise concentrated on high modes produces an enhanced dissipation term (3ν/5)Δ, with strong L^p(Ω;X) convergence to the deterministic limit (Theorem 1.3), plus a quantitative rate in L²([0,T];H^{1-δ}) (Theorem 4.2); and (iii) a uniform large deviation principle for the hyperviscous case Λ ∈ (1,2), δ = 0, with rate function given by a skeleton-equation variational formula (Theorem 1.6). The proofs use Galerkin approximations, compactness, Grönwall arguments, and the Budhiraja-Dupuis weak convergence method.

Significance. The results are a meaningful contribution to the literature on transport-noise regularization of 3D fluid equations. The paper extends the globally modified Navier-Stokes well-posedness theory to a cut-off in the H^{1-δ} norm, and it transfers the now-standard high-mode-noise scaling limit to a globally modified model, including a quantitative rate. The LDP is a nontrivial application of the weak convergence method with a concrete rate function. The proofs are largely detailed and exploit the correct quantitative tools, notably Theorem 2.2 from Luo-Tang. However, the LDP statement as written overclaims the range of r for which the skeleton equation is proved well posed; this must be fixed before the result can be accepted.

major comments (2)
  1. [Theorem 1.6 and Lemma 5.5] Theorem 1.6 states the uniform LDP for every r ∈ (0,3/2), but Lemma 5.5 proves well-posedness of the skeleton equation only under the additional condition Λ + r > 5/2. In Step 1 of Lemma 5.5, the verification of (5.2) uses the bound ‖Π(g·∇ξ)‖_{H^{-Λ}} ≲ ‖g‖_{H^r}‖ξ‖_{H^{5/2-Λ-r}} and then requires the exponent 5/2-Λ-r to be negative so that ‖ξ‖_{H^{5/2-Λ-r}} ≤ ‖ξ‖_{L²}. For example, when Λ = 1.2 and r = 0.5, the exponent is positive and the resulting integral is not controlled by the available bounds g ∈ L²H^r, ξ ∈ L∞L² ∩ L²H^Λ. Thus uniqueness of the skeleton equation is not established in this regime, and since the rate function I_{u0}(u) = inf_{g: u = G0(u0, Int(g))} (1/2)∫‖g‖²_{H^r}dt presupposes a single-valued solution map G0, the LDP is unjustified for the full stated range of r. The same condition is used in Lemma 5.6 and Lemma 5.8. The theorem should either be restricted to Λ + r > 5/2 or be supplemented by a new well-posedness proof for the remaining parameter range.
  2. [Section 3 (proof of Theorem 1.2)] The existence proof is carried out only for Λ = 1 and δ ∈ (0,1/4), and the opening paragraph of Section 3 asserts that this is enough because dissipation is enhanced for larger Λ. However, Theorem 1.2 also covers the case Λ ∈ (1,2), δ = 0, which is exactly the setting used in Theorem 1.6 and Lemma 5.8. For δ = 0 the cut-off is FN(‖u‖_{H^1}) and the compactness target is L²H^1 rather than L²H^{1-δ}; one needs Theorem 2.5(i') and a separate convergence argument for the nonlinear term. Remark 3.5 correctly notes the failure of the argument in the case Λ = 1, δ = 0, but the corresponding hyperviscous extension is not written out. Since this part of Theorem 1.2 underpins the LDP, the extension should be provided or at least sketched in sufficient detail to be checked.
minor comments (4)
  1. [Remark 1.4 and Theorem 4.2] The displayed convergence rate contains the factor ν^{δ/2+2δ}, but the derivation in the proof of Theorem 4.2, after dividing (4.8) by ν^{-2}N²‖u0‖²_{H¹}, gives ν^{δ/2+2} δ^{(δ-5)/5} T^{(7δ-δ²)/10} N^{-2}‖θn‖^{2δ/5}_{ℓ∞}. Please correct the exponent of ν in both statements.
  2. [Section 5.2 (Definition 5.4 and Theorem 1.6)] Once the condition Λ + r > 5/2 is added to Theorem 1.6, the text surrounding Definition 5.4 and the definition of S_M(H^r) should explicitly state that g is divergence-free and that the parameter r satisfies the same restriction, since the proofs of Lemmas 5.5, 5.6 and 5.8 all rely on this condition.
  3. [Lemma 3.2] In the estimate for the term I4, the proof gives a bound involving ‖θ‖²_{ℓ∞}, while the displayed text writes ‖θ‖²_{ℓ²}; the bound is valid under the standing assumption ‖θ‖_{ℓ²}=1, but the notation is a bit misleading and should be harmonized.
  4. [Remark 1.4] The phrase 'ν ≫ 1 being fixed' is confusing: ν is the noise intensity appearing in (1.2), while Theorem 4.2 treats ν as a parameter that is chosen large. Please clarify the role of ν in the convergence rate statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the scaling limit and LDP proofs use independent cited estimates and standard weak-convergence methods; the main theorem contains a separate parameter-range correctness gap, not circularity.

full rationale

The paper's central claims do not reduce to their inputs. Theorem 1.2 is proved by Galerkin approximation, compactness and an Ito formula energy estimate; the uniqueness proof follows Romito's method and does not presuppose the conclusion. Theorem 1.3 uses Theorem 2.2, cited from [40], to identify the limit of the Stratonovich-Ito corrector S_theta_n with (3nu/5)Delta. Although [40] is coauthored by the second author, the cited statement is a parameter-free analytic estimate with stated assumptions and does not assume the convergence of the stochastic GMNSE solutions, so it is independent evidence under the reviewing rules. The LDP in Theorem 1.6 is obtained through the Budhiraja-Dupuis-Maroulas weak convergence method: the rate function is defined through the skeleton solution map G0, whose well-posedness is proved in Lemma 5.5, and Hypothesis 5.3 is verified in Lemmas 5.7 and 5.8. No fitted parameter is renamed as a prediction, and no ansatz is imported solely by self-citation. An explicit, honest limitation is present: Theorem 1.6 states r in (0,3/2), while Lemma 5.5 and Lemma 5.6 impose the additional condition Lambda+r>5/2. For example, Lambda=1.2 and r=0.5 violate this condition, and the estimate ||Pi(g·grad xi)||_{H^{-Lambda}} <= ||g||_{H^r} ||xi||_{H^{5/2-Lambda-r}} used for the skeleton uniqueness is then not bounded by the available a priori estimates. This is a correctness/completeness gap in the stated parameter range, not a circular reduction, and it does not affect the circularity score.

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

No free parameters are fitted to data; the model parameters N, ν, δ, Λ, r are part of the equation. There are no invented physical entities. The main nonstandard input is the transport noise structure and the cut-off function, which are domain assumptions. The proof relies on standard analytic inequalities, compactness theorems, and stochastic calculus results, all explicitly cited.

assumptions (5)
  • domain assumption The noise coefficients {σ_{k,i}} form a complete orthonormal system and the identity S_θ(u) → (3ν/5)Δu as n→∞ (Theorem 2.2 from [40]).
    The specific form of the transport noise and the convergence of the Stratonovich-Itô corrector to (3ν/5)Δ is imported from Luo-Tang [40]; the scaling limit conclusion depends on it.
  • standard math The cut-off function F_N satisfies Lipschitz-type estimates (Lemma 3.6) and the product estimates in Lemma 3.7 hold.
    These are analytic estimates proved in the paper using standard Sobolev embedding and interpolation; they are used throughout the well-posedness and convergence proofs.
  • standard math The compact embedding theorems (Theorem 2.5) from Simon [50] hold for the relevant spaces.
    Used for tightness and existence via Skorohod representation; cited from the literature.
  • domain assumption The Itô formula and energy equality apply to the weak solutions.
    The authors invoke Rozovsky-Lototsky [48, Theorem 2.12/2.13] to justify the energy estimates; the regularity conditions needed are verified in the paper.
  • domain assumption For the LDP, the skeleton equation is well-posed when Λ + r > 5/2.
    This condition is stated and proved in Lemma 5.5, but it is omitted from the statement of Theorem 1.6; it is load-bearing for the LDP's rate function.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Scaling Limit and Large Deviation for 3D Globally Modified Stochastic Navier-Stokes Equations with Transport Noise." pith.science (2026). https://pith.science/paper/DYWZYY33

@misc{pith2026241220752,
  author       = {Pith},
  title        = {Pith review of: Scaling Limit and Large Deviation for 3D Globally Modified Stochastic Navier-Stokes Equations with Transport Noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DYWZYY33}},
  note         = {Machine review of arXiv:2412.20752}
}
read the original abstract

We consider the globally modified stochastic (hyperviscous) Navier-Stokes equations with transport noise on 3D torus. We first establish the existence and pathwise uniqueness of the weak solutions, and then show their convergence to the solutions of the deterministic 3D globally modified (hyperviscous) Navier-Stokes equations in an appropriate scaling limit. Furthermore, we prove a large deviation principle for the stochastic globally modified hyperviscous system.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. A scaling limit for Vlasov equations with electrostatic fluctuations

    math.PR 2025-07 conditional novelty 6.0 of 10

    Random electrostatic fluctuations with fixed intensity and vanishing spatial correlation produce velocity diffusion in the Vlasov-Poisson limit, proving a deterministic Vlasov-Fokker-Planck equation.

Reference graph

Works this paper leans on

54 extracted references · 51 canonical work pages · cited by 1 Pith paper

  1. [1]

    A. Agresti. Delayed blow-up and enhanced diffusion by tran sport noise for systems of reaction-diffusion equations. Stoch. Partial Differ. Equ. Anal. Comput. 12 (2024), no. 3, 1907–1981

    work page 2024

  2. [2]

    A. Agresti. Global smooth solutions by transport noise o f 3D Navier-Stokes equations with small hyperviscosity. arXiv:2406.09267v2

    arXiv

  3. [3]

    Bahouri, J

    H. Bahouri, J. Y. Chemin, R. Danchin. Fourier analysis an d nonlinear partial differential equations, Grundlehren der mathematischen Wissenschafte n, 343, Springer, Heidelberg, 2011

    work page 2011

  4. [4]

    Boussinesq

    J. Boussinesq. Essai sur la th´ eorie des eaux courantes, M´ emoires pr´ esent´ es par divers savants ` a l’Acad´ emie des Sciences XXIII, Imprimerie Nationale, Paris, 1877, pp. 1–680

    work page

  5. [5]

    Brze´ zniak, M

    Z. Brze´ zniak, M. Capi´ nski, F. Flandoli. Stochastic Na vier-Stokes equations with multi- plicative noise. Stochastic Anal. Appl. 10 (1992), no. 5, 523–532

    work page 1992

  6. [6]

    Brze´ zniak, F

    Z. Brze´ zniak, F. Flandoli, M. Maurelli. Existence and u niqueness for stochastic 2D Euler flows with bounded vorticity. Arch. Ration. Mech. Anal. 221 (2016), no. 1, 107–142. 33

    work page 2016

  7. [7]

    Budhiraja, P

    A. Budhiraja, P. Dupuis. A variational representation f or positive functionals of infinite di- mensional Brownian motion. Probab. Math. Statist. 20 (2000), no. 1, Acta Univ. Wratislav. No. 2246, 39–61

    work page 2000

  8. [8]

    Budhiraja, P

    A. Budhiraja, P. Dupuis, V. Maroulas. Large deviations f or infinite dimensional stochastic dynamical systems. Ann. Probab. 36 (2008), no. 4, 1390–1420

    work page 2008

Show all 54 references
  1. [9]

    Butori, F

    F. Butori, F. Flandoli, E. Luongo. On the Itˆ o-Stratonov ich Diffusion Limit for the Mag- netic Field in a 3D Thin Domain. arXiv:2401.15701

    arXiv

  2. [10]

    Butori, F

    F. Butori, F. Flandoli, E. Luongo, Y. Tahraoui. Backgro und Vlasov equations and Young measures for passive scalar and vector advection equations under special stochastic scaling limits. arXiv:2407.10594

    arXiv

  3. [11]

    Butori, E

    F. Butori, E. Luongo. Mean-Field Magnetohydrodynamic s Models as Scaling Limits of Stochastic Induction Equations. arXiv:2406.07206

    arXiv

  4. [12]

    Cheskidov, D.D

    A. Cheskidov, D.D. Holm, E. Olson, E.S. Titi. On a Leray- α model of turbulence. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), no. 2055, 629–649

    2005

  5. [13]

    Chueshov, A

    I. Chueshov, A. Millet. Stochastic 2D hydrodynamical t ype systems: well posedness and large deviations. Appl. Math. Optim. 61 (2010), no. 3, 379–420

    2010

  6. [14]

    Carigi, E

    G. Carigi, E. Luongo. Dissipation properties of transp ort noise in the two-layer quasi- geostrophic model. J. Math. Fluid Mech. 25 (2023), no. 2, Paper No. 28, 27 pp

    2023

  7. [15]

    Cerrai, A

    S. Cerrai, A. Debussche. Large deviations for the two-d imensional stochastic Navier-Stokes equation with vanishing noise correlation. Ann. Inst. Henri Poincar´ e Probab. Stat. 55 (2019), no. 1, 211–236

    2019

  8. [16]

    Caraballo, P

    T. Caraballo, P. E. Kloeden, J. Real. Unique strong solu tions and V-attractors of a three dimensional system of globally modified Navier-Stokes equa tions. Adv. Nonlinear Stud. 10 (2010), no. 1, 245–247

    2010

  9. [17]

    Caraballo, P

    T. Caraballo, P. E. Kloeden, J. Real. Invariant measure s and statistical solutions of the globally modified Navier-Stokes equations. Discrete Contin. Dyn. Syst. Ser. B. 10 (2008), no. 4, 761–781

    2008

  10. [18]

    Da Prato

    G. Da Prato. Kolmogorov equations for stochastic PDEs. Springer Science & Business Media, 2004

    2004

  11. [19]

    Delarue, F

    F. Delarue, F. Flandoli, D. Vincenzi. Noise prevents co llapse of Vlasov-Poisson point charges. Comm. Pure Appl. Math. 67 (2014), no. 10, 1700–1736

    2014

  12. [20]

    Flandoli, L

    F. Flandoli, L. Galeati, D. Luo. Scaling limit of stocha stic 2D Euler equations with trans- port noises to the deterministic Navier-Stokes equations. J. Evol. Equ. 21 (2021), no. 1, 567–600

    2021

  13. [21]

    Flandoli, L

    F. Flandoli, L. Galeati, D. Luo. Quantitative converge nce rates for scaling limit of SPDEs with transport noise. J. Differential Equations 394 (2024), 237–277

    2024

  14. [22]

    Flandoli, M

    F. Flandoli, M. Gubinelli, E. Priola. Well-posedness o f the transport equation by stochastic perturbation. Invent. Math. 180 (2010), no. 1, 1–53

    2010

  15. [23]

    Flandoli, M

    F. Flandoli, M. Gubinelli, E. Priola. Full well-posedn ess of point vortex dynamics corre- sponding to stochastic 2D Euler equations. Stochastic Process. Appl. 121 (2011), no. 7, 1445–1463

    2011

  16. [24]

    Flandoli, M

    F. Flandoli, M. Hofmanov´ a, D. Luo, T. Nilssen. Global w ell-posedness of the 3D Navier- Stokes equations perturbed by a deterministic vector field. Ann. Appl. Probab. 32 (2022), no. 4, 2568–2586. 34

    2022

  17. [25]

    Flandoli, D

    F. Flandoli, D. Luo. High mode transport noise improves vorticity blow-up control in 3D Navier–Stokes equations. Probab. Theory Related Fields 180 (2021), no. 1-2, 309–363

    2021

  18. [26]

    Flandoli, D

    F. Flandoli, D. Luo. On the Boussinesq hypothesis for a s tochastic Proudman-Taylor model. SIAM J. Math. Anal. 56 (2024), no. 3, 3886–3923

    2024

  19. [27]

    Flandoli, D

    F. Flandoli, D. Luo, E. Luongo. 2D Smagorinsky-type lar ge eddy models as limits of stochastic PDEs. J. Nonlinear Sci. 34 (2024), no. 3, Paper No. 54, 25 pp

    2024

  20. [28]

    Flandoli, E

    F. Flandoli, E. Luongo. Stochastic partial differential equations in fluid mechanics. Lecture Notes in Mathematics, 2330. Springer, Singapore, 2023

    2023

  21. [29]

    Flandoli, U

    F. Flandoli, U. Pappalettera. 2D Euler equations with S tratonovich transport noise as a large-scale stochastic model reduction. J. Nonlinear Sci. 31 (2021), no. 1, Paper No. 24, 38 pp

    2021

  22. [30]

    Flandoli, U

    F. Flandoli, U. Pappalettera. From additive to transpo rt noise in 2D fluid dynamics. Stoch. Partial Differ. Equ. Anal. Comput. 10 (2022), no. 3, 964–1004

    2022

  23. [31]

    L. Galeati. On the convergence of stochastic transport equations to a deterministic parabolic one. Stoch. Partial Differ. Equ. Anal. Comput. 8 (2020), no. 4, 833–868

    2020

  24. [32]

    Galeati, D

    L. Galeati, D. Luo. LDP and CLT for SPDEs with transport n oise. Stoch. Partial Differ. Equ. Anal. Comput. 12 (2024), no. 1, 736–793

    2024

  25. [33]

    D. D. Holm. Variational principles for stochastic fluid dynamics. Proc. A. 471 (2015), no. 2176, 20140963, 19 pp

    2015

  26. [34]

    P. E. Kloeden, J. A. Langa, J. Real. Pullback V -attractors of the 3-dimensional globally modified Navier-Stokes equations. Commun. Pure Appl. Anal. 6 (2007), no. 4, 937–955

    2007

  27. [35]

    Kuksin, A

    S. Kuksin, A. Shirikyan. Mathematics of two-dimension al turbulence. Cambridge Tracts in Mathematics, 194, Cambridge Univ. Press, Cambridge, 201 2

  28. [36]

    O. Lang, D. Crisan. Well-posedness for a stochastic 2D E uler equation with transport noise. Stoch. Partial Differ. Equ. Anal. Comput. 11 (2023), no. 2, 433–480

    2023

  29. [37]

    D. Luo. Absolute continuity under flows generated by SDE with measurable drift coeffi- cients. Stochastic process. Appl. 121 (2011), no. 10, 2393–2415

    2011

  30. [38]

    D. Luo. Convergence of stochastic 2D inviscid Boussine sq equations with transport noise to a deterministic viscous system. Nonlinearity 34 (2021), no. 12, 8311–8330

    2021

  31. [39]

    D. Luo. Enhanced dissipation for stochastic Navier-St okes equations with transport noise. J. Dynam. Differential Equations (2023). https://doi.org/10.1007/s10884-023-10307-w

    2023 doi

  32. [40]

    D. Luo, B. Tang. Stochastic inviscid Leray- α model with transport noise: convergence rates and CLT. Nonlinear Anal. 234 (2023), Paper No. 113301, 34 pp

    2023

  33. [41]

    T. Lange. Regularization by Noise of an Averaged Versio n of the Navier-Stokes Equations. J. Dynam. Differential Equations 36 (2024), no. 4, 3011–3036

    2024

  34. [42]

    Mikulevicius, B

    R. Mikulevicius, B. L. Rozovskii. Stochastic Navier-S tokes equations for turbulent flows. SIAM J. Math. Anal. 35 (2004), no. 5, 1250–1310

    2004

  35. [43]

    Mikulevicius, B

    R. Mikulevicius, B. L. Rozovskii. Global L2-solutions of stochastic Navier-Stokes equa- tions. Ann. Probab. 33 (2005), no. 1, 137–176

    2005

  36. [44]

    Olson, E.S

    E. Olson, E.S. Titi. Viscosity versus vorticity stretc hing: global well-posedness for a family of Navier-Stokes-alpha-like models. Nonlinear Anal. 66 (2007), no. 11, 2427–2458

    2007

  37. [45]

    Papini, F

    A. Papini, F. Flandoli, R. J. Huang. Turbulence enhance ment of coagulation: the role of eddy diffusion in velocity. Phys. D. 448 (2023), Paper No. 133726, 16 pp. 35

    2023

  38. [46]

    Z. Qiu, C. Sun. Stochastic Landau-Lifshitz-Bloch equa tion with transport noise: well- posedness, dissipation enhancement. J. Stat. Phys. 191 (2024), no. 4, Paper No. 43, 29 pp

    2024

  39. [47]

    R¨ ockner, X

    M. R¨ ockner, X. Zhang. Stochastic tamed 3D Navier-Stok es equations: existence, unique- ness and ergodicity. Probab. Theory Related Fields 145 (2009), no. 1–2, 211–267

    2009

  40. [48]

    B. L. Rozovsky, S.V. Lototsky. Stochastic Evolution Sy stems: Linear Theory and Applica- tions to Non-Linear Filtering. Probability Theory and Stoc hastic Modelling, 89, Springer, Cham, 2018

    2018

  41. [49]

    M. Romito. The uniqueness of weak solutions of the globa lly modified Navier-Stokes equa- tions. Adv. Nonlinear Stud. 9 (2009), no. 2, 425–427

    2009

  42. [50]

    J. Simon. Compact sets in the space Lp(0, T ; B). Ann. Mat. Pura Appl. (4) 146 (1987), 65–96

    1987

  43. [51]

    R. Temam. Navier-Stokes equations: theory and numeric al analysis. American Mathemat- ical Society, 2024

    2024

  44. [52]

    S. R. S. Varadhan. Large Deviations and Applications. C BMS-NSF Regional Conference Series in Applied Mathematics, 46, SIAM, Philadelphia, PA, 1984

    1984

  45. [53]

    X. Xie, F. Gao. The Delayed Effect of Multiplicative Noise on the Blow-Up for a Class of Fractional Stochastic Differential Equations. Fractal Fract. 2024, 8(3), 127

    2024

  46. [54]

    Zhang, J

    J. Zhang, J. Huang. Convergence rates and central limit theorem for 3-D stochastic frac- tional Boussinesq equations with transport noise. Phys. D. 470 (2024), 134406. 36

    2024

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.