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High mode transport noise improves vorticity blow-up control in 3D Navier–Stokes equations

Abstract

The paper is concerned with the problem of regularization by noise of 3D Navier–Stokes equations. As opposed to several attempts made with additive noise which remained inconclusive, we show here that a suitable multiplicative noise of transport type has a regularizing effect. It is proven that stochastic transport noise provides a bound on vorticity which gives well posedness, with high probability. The result holds for sufficiently large noise intensity and sufficiently high spectrum of the noise.

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References

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Babin, A., Mahalov, A., Nicolaenko, B.: Global splitting, integrability and regularity of 3D Euler and Navier–Stokes equations for uniformly rotating fluids. Eur. J. Mech. B/Fluids 15, 291–300 (1996)

    MATH  Google Scholar 

  4. Barbato, D., Bessaih, H., Ferrario, B.: On a stochastic Leray-\(\alpha \) model of Euler equations. Stoch. Proc. Appl. 124(1), 199–219 (2014)

    Article  MathSciNet  Google Scholar 

  5. Billingsley, P.: Convergence of Probability Measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. Wiley, New York (1999)

  6. Brzeźniak, Z., Capiński, M., Flandoli, F.: Stochastic Navier–Stokes equations with multiplicative noise. Stoch. Anal. Appl. 10(5), 523–532 (1992)

    Article  MathSciNet  Google Scholar 

  7. Butkovski, O., Mytnik, L.: Regularization by noise and flows of solutions for a stochastic heat equation. Ann. Probab. 47, 165–212 (2019)

    MathSciNet  MATH  Google Scholar 

  8. Chiodaroli, E., Feireisl, E., Flandoli, F.: Ill posedness for the full Euler system driven by multiplicative white noise. to appear on Indiana Univ. Math. J., see arXiv:1904.07977

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Da Prato, G., Flandoli, F.: Pathwise uniqueness for a class of SDE in Hilbert spaces and applications. J. Funct. Anal. 259, 243–267 (2010)

    Article  MathSciNet  Google Scholar 

  12. Da Prato, G., Flandoli, F., Priola, E., Röckner, M.: Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift. Ann. Probab. 41, 3306–3344 (2013)

    Article  MathSciNet  Google Scholar 

  13. Da Prato, G., Flandoli, F., Röckner, M., Veretennikov, AYu.: Strong uniqueness for SDEs in Hilbert spaces with nonregular drift. Ann. Probab. 44, 1985–2023 (2016)

    Article  MathSciNet  Google Scholar 

  14. Davie, A.M.: Uniqueness of solutions of stochastic differential equations. Int. Math. Res. Not. 24, Article ID rnm 124, 26 p. (2007)

  15. Debussche, A., Tsutsumi, Y.: 1D quintic nonlinear Schrödinger equation with white noise dispersion. J. Math. Pures Appl. 96(4), 363–376 (2011)

    Article  MathSciNet  Google Scholar 

  16. Delarue, F., Flandoli, F., Vincenzi, D.: Noise prevents collapse of Vlasov–Poisson point charges. Commun. Pures Appl. Math. 67, 1700–1736 (2014)

    Article  MathSciNet  Google Scholar 

  17. Fefferman, C.L.: Existence and smoothness of the Navier–Stokes equations, the millennium prize problems, pp. 57–67. Clay Math. Inst, Cambridge (2006)

  18. Flandoli, F., Galeati, L., Luo, D.: Scaling limit of stochastic 2D Euler equations with transport noises to the deterministic Navier–Stokes equations. J. Evol. Equ. (2020), http://link.springer.com/article/10.1007/s00028-020-00592-z

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

    Article  MathSciNet  Google Scholar 

  20. Flandoli, F., Gubinelli, M., Priola, E.: Full well-posedness of point vortex dynamics corresponding to stochastic 2D Euler equations. Stoch. Proc. Appl. 121(7), 1445–1463 (2011)

    Article  MathSciNet  Google Scholar 

  21. Flandoli, F., Luo, D.: Convergence of transport noise to Ornstein–Uhlenbeck for 2D Euler equations under the enstrophy measure. Ann. Probab. 48(1), 264–295 (2020)

    Article  MathSciNet  Google Scholar 

  22. Flandoli, F., Mahalov, A.: Stochastic three-dimensional rotating Navier–Stokes equations: averaging, convergence and regularity. Arch. Rational Mech. Anal. 205(1), 195–237 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  25. Galeati, L.: On the convergence of stochastic transport equations to a deterministic parabolic one. Stoch. Partial Differ. Equ. Anal. Comput. 8(4), 833–868 (2020)

    MathSciNet  MATH  Google Scholar 

  26. Gassiat, P., Gess, B.: Regularization by noise for stochastic Hamilton–Jacobi equations. Probab. Theory Relat. Fields 173, 1063–1098 (2019)

    Article  MathSciNet  Google Scholar 

  27. Gess, B., Maurelli, M.: Well-posedness by noise for scalar conservation laws. Commun. Partial Differ. Equ. 43(12), 1702–1736 (2018)

    Article  MathSciNet  Google Scholar 

  28. Gyöngy, I.: Existence and uniqueness results for semilinear stochastic partial differential equations. Stoch. Process. Appl. 73(2), 271–299 (1998)

    Article  MathSciNet  Google Scholar 

  29. Hofmanová, M., Zhu, R., Zhu, X.: Non-uniqueness in law of stochastic 3D Navier–Stokes equations. arXiv:1912.11841v1

  30. Holm, D.D.: Variational principles for stochastic fluid dynamics. Proc. R. Soc. A 471, 20140963 (2015)

    Article  MathSciNet  Google Scholar 

  31. Iyer, G., Xu, X., Zlatos, A.: Convection induced singularity suppression in the Keller–Siegel and other non-liner PDEs. Trans. Amer. Math. Soc. https://doi.org/10.1090/tran/8195

  32. Kunita, H.: Stochastic differential equations and stochastic flows of diffeomorphisms, Ecole d’été de probabilités de Saint-Flour, XII—1982, 143-303, Lecture Notes in Math. 1097, Springer, Berlin (1984)

  33. Krylov, N.V., Röckner, M.: Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Related Fields 131, 154–196 (2005)

    Article  MathSciNet  Google Scholar 

  34. Kurtz, T.: The Yamada–Watanabe–Engelbert theorem for general stochastic equations and inequalities. Electron. J. Probab. 12, 951–965 (2007)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  36. Mikulevicius, R., Rozovskii, B.L.: Global \(L^2\)-solutions of stochastic Navier–Stokes equations. Ann. Probab. 33(1), 137–176 (2005)

    Article  MathSciNet  Google Scholar 

  37. Rozovsky, B.L., Lototsky, S.V.: Stochastic evolution systems. Linear theory and applications to non-linear filtering. Second edition. Probability Theory and Stochastic Modelling, 89. Springer, Cham (2018)

  38. Simon, J.: Compact sets in the space \(L^p(0, T; B)\). Ann. Mat. Pura Appl. 146, 65–96 (1987)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  40. Temam, R.: Navier–Stokes Equations and Nonlinear Functional Analysis. Second edition. CBMS-NSF Regional Conference Series in Applied Mathematics, 66. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1995)

  41. Veretennikov, Y.A.: On strong solution and explicit formulas for solutions of stochastic integral equations. Math. USSR Sb. 39, 387–403 (1981)

    Article  Google Scholar 

  42. Vincent, A., Meneguzzi, M.: The spatial structure and statistical properties of homogeneous turbulence. J. Fluid Mech. 225, 1–20 (1991)

    Article  Google Scholar 

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Acknowledgements

Marek Capiński gave a talk in Ludwig Arnold group in Bremen around 1987, conjecturing that stochastic transport in parabolic PDEs could have a similar stabilizing effect as the skew-symmetric linear state dependent noise used by Arnold, Crauel and Wishtutz in their theory of stabilization by noise [1, 2]. The first author, attending that talk, was permanently inspired by that conjecture, which however is still unproven, although related results exist in many directions (see [9] and references therein). We are grateful also to Zdzisław Brzeźniak for several discussions on Capiński’s suggestion. See [6] for a first attempt to use transport noise in 2D Navier–Stokes equations. The result here is not a solution of that problem but it is based on a similar intuition. The second author is grateful to the National Key R&D Program of China (No. 2020YFA0712700), the National Natural Science Foundation of China (Nos. 11688101, 11931004, 12090014) and the Youth Innovation Promotion Association, CAS (2017003).

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Correspondence to Dejun Luo.

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Appendices

Appendix 1: Convergence of \(S_{\theta ^N}(v)\)

Recall the definition (1.6) of \(S_\theta (v)\) in the introduction. The purpose of this section is to prove

Theorem 5.1

Assume \(\theta ^N\) is given as in (1.10). Then for any smooth divergence free vector field \(v:{\mathbb {T}}^3 \rightarrow {\mathbb {R}}^3\), the following limit holds in \(L^2({\mathbb {T}}^3,{\mathbb {R}}^3)\):

$$\begin{aligned} \lim _{N\rightarrow \infty } S_{\theta ^N}(v) = \frac{3\nu }{5} \Delta v. \end{aligned}$$

First, thanks to the equality (2.7), it is sufficient to prove that, under the conditions of Theorem 5.1,

$$\begin{aligned} \lim _{N\rightarrow \infty } S_{\theta ^N}^\perp (v) = \frac{2\nu }{5} \Delta v \quad \text{ holds } \text{ in } L^2({\mathbb {T}}^3,{\mathbb {R}}^3), \end{aligned}$$
(5.1)

where the operator \(S_{\theta ^N}^\perp \) is defined in (2.6) (replacing \(\theta \) by \(\theta ^N\)). The reason for turning to the new quantity \(S_\theta ^\perp (v)\) is that we have simpler formulae for the operator \(\Pi ^\perp \) which is orthogonal to the Leray projection \(\Pi \). If X is a general vector field, then, formally,

$$\begin{aligned} \Pi ^\perp X = \nabla \Delta ^{-1} \mathrm{div}(X). \end{aligned}$$
(5.2)

On the other hand, if \(X= \sum _{l\in {\mathbb {Z}}^3_0} X_l e_l\), \(X_l\in {\mathbb {C}}^3\), then

$$\begin{aligned} \Pi ^\perp X= \sum _l \frac{l\cdot X_l}{|l|^2} l e_l = \nabla \bigg [ \frac{1}{2\pi \mathrm{i}} \sum _l \frac{l\cdot X_l}{|l|^2} e_l \bigg ]. \end{aligned}$$
(5.3)

Now we assume the divergence free vector field v has the Fourier expansion

$$\begin{aligned} v= \sum _{l,\beta } v_{l,\beta } \sigma _{l,\beta }. \end{aligned}$$

The coefficients \(\{v_{l,\beta }: l\in {\mathbb {Z}}^3_0, \beta =1,2 \} \subset {\mathbb {C}}\) satisfy \(\overline{v_{l,\beta }}= v_{-l,\beta }\). Indeed, the computations below do not require that v is a real vector field.

Lemma 5.2

We have

$$\begin{aligned} S_\theta ^\perp (v)= - \frac{6\pi ^2 \nu }{\Vert \theta \Vert _{\ell ^2}^2} \sum _{l,\beta } v_{l,\beta } \Pi \bigg \{ \bigg [ \sum _{k,\alpha } \theta _k^2 (a_{k,\alpha } \cdot l)^2 (a_{l,\beta }\cdot (k-l)) \frac{k-l}{|k-l|^2} \bigg ] e_l \bigg \}.\nonumber \\ \end{aligned}$$
(5.4)

Proof

We give two different proofs, using respectively (5.3) and (5.2).

(1) We have

$$\begin{aligned} \nabla v(x)= \sum _{l,\beta } v_{l,\beta } \nabla \sigma _{l,\beta }(x) = 2\pi \mathrm{i} \sum _{l,\beta } v_{l,\beta } (a_{l,\beta } \otimes l) e_l(x). \end{aligned}$$

Note that \(\sigma _{-k,\alpha }(x)= a_{k,\alpha } e_{-k}(x)\); thus

$$\begin{aligned} (\sigma _{-k,\alpha }\cdot \nabla v)(x) = 2\pi \mathrm{i} \sum _{l,\beta } v_{l,\beta } (a_{k,\alpha } \cdot l) a_{l,\beta } e_{l-k}(x). \end{aligned}$$

By the first equality in (5.3) and using \(a_{l,\beta } \cdot l=0\), we have

$$\begin{aligned} \begin{aligned} \Pi ^\perp (\sigma _{-k,\alpha }\cdot \nabla v)(x)&= 2\pi \mathrm{i} \sum _{l,\beta } v_{l,\beta } (a_{k,\alpha } \cdot l) (a_{l,\beta } \cdot (l-k)) \frac{l-k}{|l-k|^2} e_{l-k}(x)\\&= - 2\pi \mathrm{i} \sum _{l,\beta } v_{l,\beta } (a_{k,\alpha } \cdot l) (a_{l,\beta }\cdot k) \frac{l-k}{|l-k|^2} e_{l-k}(x). \end{aligned} \end{aligned}$$
(5.5)

As a consequence,

$$\begin{aligned} \begin{aligned}&\big [ \sigma _{k,\alpha }\cdot \nabla \Pi ^\perp (\sigma _{-k,\alpha }\cdot \nabla v) \big ](x)\\&\quad = - 2\pi \mathrm{i} \sum _{l,\beta } v_{l,\beta } (a_{k,\alpha } \cdot l) (a_{l,\beta }\cdot k) \frac{l-k}{|l-k|^2} e_k(x) a_{k,\alpha } \cdot \nabla e_{l-k}(x) \\&\quad = - (2\pi \mathrm{i})^2 \sum _{l,\beta } v_{l,\beta } (a_{k,\alpha } \cdot l) (a_{l,\beta }\cdot k) \frac{l-k}{|l-k|^2} (a_{k,\alpha } \cdot (l-k)) e_k(x) e_{l-k}(x) \\&\quad = -4\pi ^2 \sum _{l,\beta } v_{l,\beta } (a_{k,\alpha } \cdot l)^2 (a_{l,\beta }\cdot k) \frac{k-l}{|k-l|^2} e_l(x). \end{aligned} \end{aligned}$$

This immediately gives us the desired identity since \(C_\nu ^2= 3\nu /2\).

(2) In the second proof we use (5.2). Since v is divergence free, we have \(\mathrm{div}(\sigma _{-k,\alpha }\cdot \nabla v)= (\nabla \sigma _{-k,\alpha }): (\nabla v)^*\), where  :  is the inner product of matrices and \(*\) means (real) transposition. Therefore,

$$\begin{aligned} \begin{aligned} \mathrm{div}(\sigma _{-k,\alpha }\cdot \nabla v)&= \big [-2\pi \mathrm{i} (a_{k,\alpha } \otimes k) e_{-k}(x)\big ] : \bigg [ 2\pi \mathrm{i} \sum _{l,\beta } v_{l,\beta } (a_{l,\beta } \otimes l) e_l(x) \bigg ]^*\\&= 4\pi ^2 \sum _{l,\beta } v_{l,\beta } \big [ (a_{k,\alpha } \otimes k): (l \otimes a_{l,\beta } )\big ] e_{l-k}(x) \\&= 4\pi ^2 \sum _{l,\beta } v_{l,\beta } (a_{k,\alpha } \cdot l) (a_{l,\beta } \cdot k) e_{l-k}(x). \end{aligned} \end{aligned}$$

This implies

$$\begin{aligned} \Delta ^{-1} \mathrm{div}(\sigma _{-k,\alpha }\cdot \nabla v)= - \sum _{l,\beta } v_{l,\beta } (a_{k,\alpha } \cdot l) (a_{l,\beta } \cdot k) \frac{e_{l-k}(x)}{|l-k|^2}, \end{aligned}$$

and thus,

$$\begin{aligned}&\Pi ^\perp (\sigma _{-k,\alpha }\cdot \nabla v)= \nabla \Delta ^{-1} \mathrm{div}(\sigma _{-k,\alpha }\cdot \nabla v)\nonumber \\&\quad = -2\pi \mathrm{i} \sum _{l,\beta } v_{l,\beta } (a_{k,\alpha } \cdot l) (a_{l,\beta } \cdot k) \frac{l-k}{|l-k|^2}e_{l-k}(x). \end{aligned}$$

This coincides with (5.5). The rest of the computations are the same as those in the first proof, so we omit them. \(\square \)

Corollary 5.3

Denote by \(\angle _{k,l}\) the angle between the vectors k and l. We have

$$\begin{aligned} S_\theta ^\perp (v)= - \frac{6\pi ^2 \nu }{\Vert \theta \Vert _{\ell ^2}^2} \sum _{l,\beta } v_{l,\beta } |l|^2 \Pi \bigg \{ \bigg [ \sum _{k} \theta _k^2 \sin ^2(\angle _{k,l}) (a_{l,\beta }\cdot (k-l)) \frac{k-l}{|k-l|^2} \bigg ] e_l \bigg \}. \end{aligned}$$

Proof

Recall that \(\{\frac{k}{|k|}, a_{k,1}, a_{k,2}\}\) is an ONS of \({\mathbb {R}}^3\); we have

$$\begin{aligned} \sum _{\alpha =1}^2 (a_{k,\alpha } \cdot l)^2 = |l|^2 - \bigg (l\cdot \frac{k}{|k|}\bigg )^2 = |l|^2 \bigg (1- \frac{(k\cdot l)^2}{|k|^2 |l|^2} \bigg ) = |l|^2 \sin ^2(\angle _{k,l}) . \end{aligned}$$

Thus,

$$\begin{aligned} \sum _{k,\alpha } \theta _k^2 (a_{k,\alpha } \cdot l)^2 (a_{l,\beta }\cdot (k-l)) \frac{k-l}{|k-l|^2} = |l|^2 \sum _{k} \theta _k^2 \sin ^2(\angle _{k,l}) (a_{l,\beta }\cdot (k-l)) \frac{k-l}{|k-l|^2}. \end{aligned}$$

Substituting this equality into (5.4) leads to the desired result. \(\square \)

Recall the sequence \(\theta ^N \in \ell ^2\) defined in (1.10). The next result is a crucial step for proving the limit (5.1).

Proposition 5.4

For any fixed \(l\in {\mathbb {Z}}^3_0\) and \(\beta \in \{1,2\}\),

$$\begin{aligned} \lim _{N\rightarrow \infty } \frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2} \sum _{k} \big (\theta ^N_k \big )^2 \sin ^2(\angle _{k,l}) (a_{l,\beta }\cdot (k-l)) \frac{k-l}{|k-l|^2} = \frac{4}{15} a_{l,\beta }. \end{aligned}$$

Suppose we have already proved this result; we now turn to prove (5.1).

Proof of (5.1)

By Corollary 5.3, for any \(N\ge 1\),

$$\begin{aligned} S_{\theta ^N}^\perp (v)= - 6\pi ^2 \nu \sum _{l,\beta } v_{l,\beta } |l|^2 \Pi \bigg \{ \bigg [ \frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2}\sum _{k} \big (\theta ^N_k \big )^2 \sin ^2(\angle _{k,l}) (a_{l,\beta }\cdot (k-l)) \frac{k-l}{|k-l|^2} \bigg ] e_l \bigg \}. \end{aligned}$$

Since

$$\begin{aligned} \frac{2\nu }{5} \Delta v= -\frac{8\pi ^2 \nu }{5} \sum _{l,\beta } v_{l,\beta } |l|^2 a_{l,\beta } e_l \end{aligned}$$

which is divergence free, we have

$$\begin{aligned} \begin{aligned}&S_{\theta ^N}^\perp (v) -\frac{2\nu }{5} \Delta v \\&\quad = - 6\pi ^2 \nu \sum _{l,\beta } v_{l,\beta } |l|^2 \Pi \bigg \{ \bigg [ \frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2}\sum _{k} \big (\theta ^N_k \big )^2 \sin ^2(\angle _{k,l}) (a_{l,\beta }\cdot (k-l)) \frac{k-l}{|k-l|^2} - \frac{4}{15} a_{l,\beta } \bigg ] e_l \bigg \} . \end{aligned} \end{aligned}$$

Fix any big \(M>0\). We have

$$\begin{aligned} \bigg \Vert S_{\theta ^N}^\perp (v) -\frac{2\nu }{5} \Delta v \bigg \Vert _{L^2} \le K_{M,1} + K_{M,2}, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} K_{M,1} \le C \sum _{|l|\le M,\beta } |v_{l,\beta }|\, |l|^2 \bigg | \frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2}\sum _{k} \big (\theta ^N_k \big )^2 \sin ^2(\angle _{k,l}) (a_{l,\beta }\cdot (k-l)) \frac{k-l}{|k-l|^2} - \frac{4}{15} a_{l,\beta } \bigg | \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} K_{M,2}&\le C \sum _{|l|> M,\beta } |v_{l,\beta }|\, |l|^2 \bigg | \frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2}\sum _{k} \big (\theta ^N_k \big )^2 \sin ^2(\angle _{k,l}) (a_{l,\beta }\cdot (k-l)) \frac{k-l}{|k-l|^2} - \frac{4}{15} a_{l,\beta } \bigg |\\&\le C \sum _{|l|> M,\beta } |v_{l,\beta }|\, |l|^2 \bigg (\frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2}\sum _{k} \big (\theta ^N_k \big )^2 + \frac{4}{15} \bigg ) \le 2C \sum _{|l|> M,\beta } |v_{l,\beta }|\, |l|^2. \end{aligned} \end{aligned}$$

Since M is fixed, Proposition 5.4 implies that \(K_{M,1}\) vanishes as \(N\rightarrow \infty \), hence

$$\begin{aligned} \limsup _{N\rightarrow \infty } \bigg \Vert S_{\theta ^N}^\perp (v) -\frac{2\nu }{5} \Delta v \bigg \Vert _{L^2} \le 2C \sum _{|l|> M,\beta } |v_{l,\beta }|\, |l|^2. \end{aligned}$$

As the vector field v is smooth, the coefficients \(v_{l,\beta }\) decrease to 0 as \(|l|\rightarrow \infty \) faster than any polynomials of negative order. Thus we complete the proof by letting \(M\rightarrow \infty \). \(\square \)

Next we prove Proposition 5.4 for which we need a simple preparation.

Lemma 5.5

Fix \(l\in {\mathbb {Z}}^3_0\). For all \(k\in {\mathbb {Z}}^3_0\) with |k| big enough, one has

$$\begin{aligned} \bigg |\frac{(k-l)\otimes (k-l)}{|k-l|^2} - \frac{k\otimes k}{|k|^2}\bigg | \le 4\frac{|l|}{|k|}. \end{aligned}$$

Proof

We have

$$\begin{aligned} \frac{(k-l)\otimes (k-l)}{|k-l|^2} - \frac{k\otimes k}{|k|^2} = \frac{k-l}{|k-l|} \otimes \bigg (\frac{k-l}{|k-l|} - \frac{k}{|k|} \bigg ) + \bigg (\frac{k-l}{|k-l|} - \frac{k}{|k|} \bigg ) \otimes \frac{k}{|k|}, \end{aligned}$$

and thus

$$\begin{aligned} \bigg |\frac{(k-l)\otimes (k-l)}{|k-l|^2} - \frac{k\otimes k}{|k|^2}\bigg | \le 2\bigg | \frac{k-l}{|k-l|} - \frac{k}{|k|} \bigg |. \end{aligned}$$

Next, since

$$\begin{aligned} \frac{k-l}{|k-l|} - \frac{k}{|k|}= \bigg (\frac{1}{|k-l|} - \frac{1}{|k|}\bigg )(k-l) - \frac{l}{|k|}, \end{aligned}$$

one has

$$\begin{aligned} \bigg | \frac{k-l}{|k-l|} - \frac{k}{|k|} \bigg | \le \frac{\big | |k| -|k-l| \big |}{ |k|} + \frac{|l|}{|k|} \le 2 \frac{|l|}{|k|}. \end{aligned}$$

Summarizing the above estimates completes the proof. \(\square \)

Now we are ready to provide the

Proof of Proposition 5.4

Note that, by Lemma 5.5,

$$\begin{aligned} \bigg |(a_{l,\beta }\cdot (k-l)) \frac{k-l}{|k-l|^2} -(a_{l,\beta }\cdot k) \frac{k}{|k|^2} \bigg | \le \bigg |\frac{(k-l)\otimes (k-l)}{|k-l|^2} - \frac{k\otimes k}{|k|^2}\bigg | \le 4\frac{|l|}{|k|} . \end{aligned}$$

Recall the definition of \(\theta ^N\) in (1.10); then

$$\begin{aligned} \begin{aligned}&\frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2} \sum _{k} \big (\theta ^N_k \big )^2 \sin ^2(\angle _{k,l}) \bigg |(a_{l,\beta }\cdot (k-l)) \frac{k-l}{|k-l|^2} -(a_{l,\beta }\cdot k) \frac{k}{|k|^2} \bigg | \\&\quad \le \frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2} \sum _{|k|\ge N} \big (\theta ^N_k \big )^2 \times 4\frac{|l|}{|k|} \le \frac{4|l|}{N} \rightarrow 0 \end{aligned} \end{aligned}$$

as \(N\rightarrow \infty \). Therefore, it is sufficient to prove

$$\begin{aligned} \lim _{N\rightarrow \infty } \frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2} \sum _{k} \big (\theta ^N_k \big )^2 \sin ^2(\angle _{k,l}) (a_{l,\beta }\cdot k) \frac{k}{|k|^2} = \frac{4}{15} a_{l,\beta }. \end{aligned}$$
(5.6)

Lemma 5.6

Let \(\theta ^N\) be given as in (1.10). We have

$$\begin{aligned} \begin{aligned}&\lim _{N\rightarrow \infty } \frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2} \sum _{k} \big (\theta ^N_k \big )^2 \sin ^2(\angle _{k,l}) (a_{l,\beta }\cdot k) \frac{k}{|k|^2} \\&\quad = \lim _{N\rightarrow \infty } \frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2} \int _{\{N\le |x|\le 2N\}} \frac{1}{|x|^{2\gamma }} \sin ^2(\angle _{x,l}) (a_{l,\beta }\cdot x) \frac{x}{|x|^2} \,\mathrm{d}x. \end{aligned} \end{aligned}$$
(5.7)

We postpone the proof of Lemma 5.6 and continue proving Proposition 5.4. Let \(J_\beta (N)\) be the quantity on the right hand side of (5.7), which is a vector in \({\mathbb {R}}^3\). To compute \(J_\beta (N)\), we consider the new coordinate system \((y_1, y_2, y_3)\) in which the coordinate axes are \(a_{l,1}, a_{l,2}\) and \(\frac{l}{|l|}\), respectively. Let U be the orthogonal transformation matrix: \(x=Uy\). For \(i\in \{1,2,3\}\), let \(\mathrm{e}_i\in {\mathbb {R}}^3\) be such that \(\mathrm{e}_{i,j}= \delta _{i,j}\), \(1\le j\le 3\). We have

$$\begin{aligned} a_{l,i}= U \mathrm{e}_i\, (i=1,2)\quad \text{ and } \quad \frac{l}{|l|} = U \mathrm{e}_3. \end{aligned}$$

Now \(\angle _{x,l} = \angle _{Uy,U\mathrm{e}_3} = \angle _{y,\mathrm{e}_3}\) and

$$\begin{aligned} \begin{aligned} J_\beta (N)&= \frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2} \int _{\{N\le |y|\le 2N\}} \frac{1}{|y|^{2\gamma }} \sin ^2(\angle _{y,\mathrm{e}_3})\, (U\mathrm{e}_\beta \cdot Uy) \frac{Uy}{|y|^2} \,\mathrm{d}y \\&= U \bigg [ \frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2} \int _{\{N\le |y|\le 2N\}} \frac{1}{|y|^{2\gamma }} \sin ^2(\angle _{y,\mathrm{e}_3})\, \frac{y_\beta y}{|y|^2} \,\mathrm{d}y \bigg ] . \end{aligned} \end{aligned}$$
(5.8)

We denote \({{\tilde{J}}}_\beta (N)\) the term in the square bracket in (5.8), i.e. \({{\tilde{J}}}_\beta (N) = U^*J_\beta (N) \in {\mathbb {R}}^3\). By symmetry argument, we see that

$$\begin{aligned} {{\tilde{J}}}_{\beta , i}(N) = \frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2} \int _{\{N\le |y|\le 2N\}} \frac{1}{|y|^{2\gamma }} \sin ^2(\angle _{y,\mathrm{e}_3})\, \frac{y_\beta y_i}{|y|^2} \,\mathrm{d}y =0, \quad i\in \{1,2,3\}\setminus \{\beta \}.\nonumber \\ \end{aligned}$$
(5.9)

This can also be directly computed by using the spherical coordinates below.

Next, we compute \({{\tilde{J}}}_{\beta ,\beta }\, (\beta =1,2)\) by changing the variables into the spherical coordinate system:

$$\begin{aligned} {\left\{ \begin{array}{ll} y_1= r\sin \psi \cos \varphi , \\ y_2 = r\sin \psi \sin \varphi , \\ y_3= r\cos \psi , \end{array}\right. } \quad N\le r\le 2N,\, 0\le \psi \le \pi , 0\le \varphi < 2\pi . \end{aligned}$$

In this system, \(\angle _{y,\mathrm{e}_3} = \psi \). We have

$$\begin{aligned} \begin{aligned} {{\tilde{J}}}_{1,1}(N)&= \frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2} \int _N^{2N} \mathrm{d}r \int _0^\pi \mathrm{d}\psi \int _0^{2\pi } \mathrm{d}\varphi \, \frac{1}{r^{2\gamma }} (\sin ^2 \psi ) (\sin \psi \cos \varphi )^2 \, r^2 \sin \psi \\&= \frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2} \int _N^{2N}\frac{\mathrm{d}r}{r^{2\gamma -2}} \int _0^\pi \sin ^5 \psi \, \mathrm{d}\psi \int _0^{2\pi } \cos ^2\varphi \, \mathrm{d}\varphi . \end{aligned} \end{aligned}$$

Note that \(\int _0^{2\pi } \cos ^2\varphi \, \mathrm{d}\varphi = \int _0^{2\pi } \frac{1}{2} (1+ \cos 2\varphi )\, \mathrm{d}\varphi = \pi \) and

$$\begin{aligned} \begin{aligned} \int _0^\pi \sin ^5 \psi \, \mathrm{d}\psi&= - \int _0^\pi (1-\cos ^2\psi )^2 \, \mathrm{d}\cos \psi = - \int _0^\pi \big (1-2\cos ^2\psi + \cos ^4\psi \big ) \, \mathrm{d}\cos \psi \\&= - \bigg (\cos \psi -\frac{2}{3} \cos ^3\psi + \frac{1}{5} \cos ^5\psi \bigg )\bigg |_0^{\pi } = \frac{16}{15}. \end{aligned} \end{aligned}$$

Thus

$$\begin{aligned} {{\tilde{J}}}_{1,1}(N) = \frac{16}{15} \pi \times \frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2} \int _N^{2N}\frac{\mathrm{d}r}{r^{2\gamma -2}} . \end{aligned}$$
(5.10)

Following the proof of Lemma 5.6 (it is much simpler here since the function g can be taken identically 1), one can show

$$\begin{aligned} \bigg |\sum _{k} \big (\theta ^N_k \big )^2 - \int _{\{N\le |x|\le 2N\}} \frac{\mathrm{d}x}{|x|^{2\gamma }} \bigg |\le \frac{C}{N} \big \Vert \theta ^N \big \Vert _{\ell ^2}^2 \end{aligned}$$
(5.11)

for some constant \(C>0\). Equivalently,

$$\begin{aligned} \bigg | \big \Vert \theta ^N \big \Vert _{\ell ^2}^2 - 4\pi \int _N^{ 2N} \frac{\mathrm{d}r}{r^{2\gamma -2}} \bigg |\le \frac{C}{N} \big \Vert \theta ^N \big \Vert _{\ell ^2}^2, \end{aligned}$$

which implies

$$\begin{aligned} \bigg | \frac{1}{4\pi } - \frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2} \int _N^{ 2N} \frac{\mathrm{d}r}{r^{2\gamma -2}} \bigg |\le \frac{C}{N }. \end{aligned}$$

Recalling (5.10), we obtain \(\lim _{N\rightarrow \infty } {{\tilde{J}}}_{1,1}(N) = \frac{4}{15}\), which, combined with (5.9), implies

$$\begin{aligned} \lim _{N\rightarrow \infty } {{\tilde{J}}}_1(N) = \frac{4}{15} \mathrm{e}_1. \end{aligned}$$

Therefore, by (5.8),

$$\begin{aligned} \lim _{N\rightarrow \infty } J_1(N) = \lim _{N\rightarrow \infty } U {{\tilde{J}}}_1(N) = \frac{4}{15} U \mathrm{e}_1 =\frac{4}{15} a_{l,1} . \end{aligned}$$

Similarly,

$$\begin{aligned} \begin{aligned} {{\tilde{J}}}_{2,2}(N)&= \frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2} \int _N^{2N} \mathrm{d}r \int _0^\pi \mathrm{d}\psi \int _0^{2\pi } \mathrm{d}\varphi \, \frac{1}{r^{2\gamma }} (\sin ^2 \psi ) (\sin \psi \sin \varphi )^2 \, r^2 \sin \psi \\&= {{\tilde{J}}}_{1,1}(N) \rightarrow \frac{4}{15}, \end{aligned} \end{aligned}$$

and thus \(\lim _{N\rightarrow \infty } {{\tilde{J}}}_2(N) = \frac{4}{15} \mathrm{e}_2\). As a result,

$$\begin{aligned} \lim _{N\rightarrow \infty } J_2(N) = \lim _{N\rightarrow \infty } U {{\tilde{J}}}_2(N) = \frac{4}{15} U \mathrm{e}_2 =\frac{4}{15} a_{l,2} . \end{aligned}$$

Combining these two results with (5.7), we obtain (5.6). \(\square \)

Now we provide the

Proof of Lemma 5.6

We define the function

$$\begin{aligned} g(x) = \sin ^2(\angle _{x,l}) (a_{l,\beta }\cdot x) \frac{x}{|x|^2}, \quad x\in {\mathbb {R}}^3,\, x\ne 0. \end{aligned}$$

Clearly, \(\Vert g\Vert _\infty \le 1\). We shall prove that

$$\begin{aligned} \begin{aligned} \bigg |\sum _{k} \big (\theta ^N_k \big )^2 g(k) -\int _{\{N\le |x|\le 2N\}} \frac{g(x)}{|x|^{2\gamma }} \,\mathrm{d}x \bigg | \le \frac{C}{N} \big \Vert \theta ^N \big \Vert _{\ell ^2}^2. \end{aligned} \end{aligned}$$
(5.12)

Let \(\square (k)\) be the unit cube centered at \(k\in {\mathbb {Z}}^3\) such that all sides have length 1 and are parallel to the axes. Note that for all \(k,l\in {\mathbb {Z}}^3\), \(k\ne l\), the interiors of \(\square (k)\) and \(\square (l)\) are disjoint. Let \(S_N = \bigcup _{N\le |k|\le 2N} \square (k)\); then,

$$\begin{aligned} \bigg |\sum _{k} \big (\theta ^N_k \big )^2 g(k) - \int _{S_N} \frac{g(x)}{|x|^{2\gamma }} \,\mathrm{d}x\bigg | \le \sum _{N\le |k|\le 2N} \int _{\square (k)} \bigg |\frac{g(k)}{|k|^{2\gamma }} - \frac{g(x)}{|x|^{2\gamma }} \bigg |\, \mathrm{d}x. \end{aligned}$$

It holds that, for all |k| big enough and \(x\in \square (k)\),

$$\begin{aligned} \bigg |\frac{g(k)}{|k|^{2\gamma }} - \frac{g(x)}{|x|^{2\gamma }} \bigg | \le \bigg |\frac{1}{|k|^{2\gamma }} - \frac{1}{|x|^{2\gamma }} \bigg | + \frac{|g(k) -g(x)|}{|x|^{2\gamma }} \le C\bigg (\frac{1}{|k|^{2\gamma +1}} + \frac{|g(k) -g(x)|}{|k|^{2\gamma }}\bigg ) . \end{aligned}$$

Next,

$$\begin{aligned} \begin{aligned} |g(k) -g(x)|&\le |\sin ^2(\angle _{k,l}) -\sin ^2(\angle _{x,l})| + \bigg | (a_{l,\beta }\cdot k) \frac{k}{|k|^2} - (a_{l,\beta }\cdot x) \frac{x}{|x|^2}\bigg | \\&\le 2|\sin (\angle _{k,l}) -\sin (\angle _{x,l})| + \bigg | \frac{k\otimes k}{|k|^2} - \frac{x\otimes x}{|x|^2}\bigg | \\&\le 2| \angle _{k,l} -\angle _{x,l}| + 2 \bigg | \frac{k}{|k|} - \frac{x}{|x|}\bigg |. \end{aligned} \end{aligned}$$

Since \(|x-k|\le 1\) and \(|k|\ge N \gg 1\), one has

$$\begin{aligned} |g(k) -g(x)| \le \frac{C}{|k|}. \end{aligned}$$

Summarizing the above discussions, we obtain

$$\begin{aligned}&\bigg |\sum _{k} \big (\theta ^N_k \big )^2 g(k) - \int _{S_N} \frac{g(x)}{|x|^{2\gamma }} \,\mathrm{d}x\bigg | \le \sum _{N\le |k|\le 2N} \int _{\square (k)} \frac{C}{|k|^{2\gamma +1}}\,\mathrm{d}x \\&\quad \le \frac{C}{N} \sum _{N\le |k|\le 2N} \frac{1}{|k|^{2\gamma }} = \frac{C}{N} \big \Vert \theta ^N \big \Vert _{\ell ^2}^2. \end{aligned}$$

Note that there is a small difference between the sets \(\{N\le |x|\le 2N\}\) and \(S_N\), but, in the same way, one can show that

$$\begin{aligned} \bigg |\int _{\{N\le |x|\le 2N\}} \frac{g(x)}{|x|^{2\gamma }} \,\mathrm{d}x - \int _{S_N} \frac{g(x)}{|x|^{2\gamma }} \,\mathrm{d}x\bigg | \le \frac{C}{N} \big \Vert \theta ^N \big \Vert _{\ell ^2}^2. \end{aligned}$$

Indeed, for any \(x\in \square (k)\) with \(N\le |k| \le 2N\), one has \(N-1 \le |x| \le 2N+1\). Therefore,

$$\begin{aligned} S_N = \bigcup _{N\le |k| \le 2N} \square (k) \subset \{N-1 \le |x| \le 2N+1 \} =: T_N. \end{aligned}$$

One also has

$$\begin{aligned} R_N:= \{N+1 \le |x| \le 2N-1 \} \subset S_N. \end{aligned}$$

Denote by \(A\Delta B\) the symmetric difference of sets \(A,B\subset {\mathbb {R}}^3\); then,

$$\begin{aligned} \begin{aligned}&\bigg |\int _{\{N\le |x|\le 2N\}} \frac{g(x)}{|x|^{2\gamma }} \,\mathrm{d}x - \int _{S_N} \frac{g(x)}{|x|^{2\gamma }} \,\mathrm{d}x\bigg | = \bigg |\int _{S_N\Delta \{N\le |x|\le 2N\}} \frac{g(x)}{|x|^{2\gamma }} \,\mathrm{d}x \bigg | \\&\quad \le \int _{S_N\Delta \{N\le |x|\le 2N\}} \frac{1}{|x|^{2\gamma }} \,\mathrm{d}x \le \int _{T_N\setminus R_N} \frac{1}{|x|^{2\gamma }} \,\mathrm{d}x \le \frac{C}{N^{2\gamma -2}} \le \frac{C}{N} \big \Vert \theta ^N \big \Vert _{\ell ^2}^2, \end{aligned} \end{aligned}$$

where the last step follows from

$$\begin{aligned} \begin{aligned} \big \Vert \theta ^N \big \Vert _{\ell ^2}^2&= \sum _{N\le |k| \le 2N} \frac{1}{|k|^{2\gamma }} \ge \frac{1}{(2N)^{2\gamma }}\, \#\{k\in {\mathbb {Z}}^3_0: N\le |k| \le 2N \} \ge \frac{C}{N^{2\gamma -3}}. \end{aligned} \end{aligned}$$

The proof is complete. \(\square \)

Remark 5.7

Assume \(\gamma \in [0,3/2]\). For any \(N\in {\mathbb {N}}\), we define

$$\begin{aligned} \theta ^N_k = \frac{1}{|k|^\gamma } \mathbf{1}_{\{|k|\le N\}}, \quad k\in {\mathbb {Z}}^3_0. \end{aligned}$$

Then \(\Vert \theta ^N\Vert _{\ell ^\infty } =1\) and \(\Vert \theta ^N\Vert _{\ell ^2} \rightarrow \infty \) as \(N\rightarrow \infty \). Thus the sequence \(\{\theta ^N \}_{N\in {\mathbb {N}}}\) satisfies the property (1.11). With suitable modifications of the proofs in this section, we can still prove Theorem 5.1. Indeed, the arguments above the proof of Proposition 5.4 remain the same. To prove Proposition 5.4, we fix \(M\in {\mathbb {N}}\); then for all \(N> M\),

$$\begin{aligned} \begin{aligned}&\frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2} \sum _{k} \big (\theta ^N_k \big )^2 \sin ^2(\angle _{k,l}) \bigg |(a_{l,\beta }\cdot (k-l)) \frac{k-l}{|k-l|^2} -(a_{l,\beta }\cdot k) \frac{k}{|k|^2} \bigg | \\&\quad \le \frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2} \sum _{|k|\le M} 2 \big (\theta ^N_k \big )^2 + \frac{1}{\Vert \theta ^N \Vert _{\ell ^2}^2} \sum _{|k|> M} \big (\theta ^N_k \big )^2 \times 4\frac{|l|}{|k|} \\&\quad \le C_M \frac{\Vert \theta ^N \Vert _{\ell ^\infty }^2}{\Vert \theta ^N \Vert _{\ell ^2}^2} + 4\frac{|l|}{M}. \end{aligned} \end{aligned}$$

First letting \(N\rightarrow \infty \) and then \(M\rightarrow \infty \) we see that it is sufficient to prove the limit (5.6). In the subsequent proofs, similar modifications work as well and we can complete the proof of Theorem 5.1.

Appendix 2: The difficulty with the advection noise

In this part we do some formal computations to illustrate why we cannot deal with 3D Navier–Stokes equations (1.1) with the full advection noise. Using our vector fields \(\{\sigma _{k,\alpha }: k\in {\mathbb {Z}}^3_0, \alpha =1,2\}\), the equations can be written as

$$\begin{aligned} \mathrm{d}\xi + {\mathcal {L}}_u \xi \,\mathrm{d}t = \Delta \xi \,\mathrm{d}t + \frac{C_\nu }{\Vert \theta \Vert _{\ell ^2}} \sum _{k,\alpha } \theta _k {\mathcal {L}}_{\sigma _{k,\alpha }} \xi \circ \mathrm{d}W^{k,\alpha }_t, \end{aligned}$$

where, as usual, u is related to \(\xi \) via the Biot–Savart law. It has the Itô formulation

$$\begin{aligned} \begin{aligned} \mathrm{d}\xi + {\mathcal {L}}_{u} \xi \,\mathrm{d}t&= \Delta \xi \,\mathrm{d}t + \frac{C_\nu }{\Vert \theta \Vert _{\ell ^2}} \sum _{k,\alpha } \theta _k {\mathcal {L}}_{\sigma _{k,\alpha }} \xi \, \mathrm{d}W^{k,\alpha }_t + \frac{C_\nu ^2}{\Vert \theta \Vert _{\ell ^2}^2} \sum _{k,\alpha } \theta _k^2 {\mathcal {L}}_{\sigma _{k,\alpha }} \big ( {\mathcal {L}}_{\sigma _{-k,\alpha }} \xi \big ) \, \mathrm{d}t. \end{aligned} \end{aligned}$$

By Proposition 6.1 below, this equation can be reduced to

$$\begin{aligned} \mathrm{d}\xi + {\mathcal {L}}_{u} \xi \,\mathrm{d}t = (1+\nu ) \Delta \xi \,\mathrm{d}t + \frac{C_\nu }{\Vert \theta \Vert _{\ell ^2}} \sum _{k,\alpha } \theta _k {\mathcal {L}}_{\sigma _{k,\alpha }} \xi \, \mathrm{d}W^{k,\alpha }_t. \end{aligned}$$
(6.1)

Proposition 6.1

It holds that

$$\begin{aligned} \sum _{k,\alpha } \theta _k^2 {\mathcal {L}}_{\sigma _{k,\alpha }} \big ( {\mathcal {L}}_{\sigma _{-k,\alpha }} \xi \big ) = \frac{2}{3}\Vert \theta \Vert _{\ell ^2} \Delta \xi . \end{aligned}$$

Proof

First, for any \(k\in {\mathbb {Z}}^3_0\), we have

$$\begin{aligned} \xi \cdot \nabla \sigma _{k,\alpha }= 2\pi \mathrm{i} (\xi \cdot k) \sigma _{k,\alpha }, \quad \alpha =1,2. \end{aligned}$$
(6.2)

Thus,

$$\begin{aligned} {\mathcal {L}}_{\sigma _{k,\alpha }} \xi = \sigma _{k,\alpha }\cdot \nabla \xi - 2\pi \mathrm{i} (k\cdot \xi ) \sigma _{k,\alpha }, \quad \alpha =1,2. \end{aligned}$$
(6.3)

Next we prove that for any \(k\in {\mathbb {Z}}^3_0\) and \(\alpha =1,2\),

$$\begin{aligned} {\mathcal {L}}_{\sigma _{k,\alpha }} \big ({\mathcal {L}}_{\sigma _{-k,\alpha }} \xi \big )= \mathrm{Tr}\big [(a_{k,\alpha } \otimes a_{k,\alpha }) \nabla ^2 \xi \big ] . \end{aligned}$$
(6.4)

The desired equality follows immediately from this fact and (2.3).

We have

$$\begin{aligned} {\mathcal {L}}_{\sigma _{k,\alpha }} \big ({\mathcal {L}}_{\sigma _{-k,\alpha }} \xi \big ) = \sigma _{k,\alpha }\cdot \nabla \big ({\mathcal {L}}_{\sigma _{-k,\alpha }} \xi \big ) - \big ({\mathcal {L}}_{\sigma _{-k,\alpha }} \xi \big )\cdot \nabla \sigma _{k,\alpha } =: I_1 -I_2 . \end{aligned}$$

By (6.3),

$$\begin{aligned} I_1= \sigma _{k,\alpha }\cdot \nabla \big ( \sigma _{-k,\alpha }\cdot \nabla \xi + 2\pi \mathrm{i} (k\cdot \xi ) \sigma _{-k,\alpha }\big ). \end{aligned}$$

The definition (2.1) of \(\sigma _{k,\alpha }\) leads to

$$\begin{aligned} \sigma _{k,\alpha }\cdot \nabla \sigma _{k,\alpha }= \sigma _{k,\alpha }\cdot \nabla \sigma _{-k,\alpha } =0, \quad k\in {\mathbb {Z}}^3_0. \end{aligned}$$
(6.5)

Therefore,

$$\begin{aligned} \begin{aligned} I_1&= \mathrm{Tr}\big [(\sigma _{k,\alpha } \otimes \sigma _{-k,\alpha }) \nabla ^2 \xi \big ]+ 2\pi \mathrm{i} \big [\sigma _{k,\alpha }\cdot \nabla (k\cdot \xi ) \big ] \sigma _{-k,\alpha } \\&= \mathrm{Tr}\big [(a_{k,\alpha } \otimes a_{k,\alpha }) \nabla ^2 \xi \big ]+ 2\pi \mathrm{i} \big [k\cdot ( a_{k,\alpha }\cdot \nabla \xi ) \big ] a_{k,\alpha }. \end{aligned} \end{aligned}$$

Next, by 6.3 and (6.5),

$$\begin{aligned} I_2 = \big ( \sigma _{-k,\alpha }\cdot \nabla \xi + 2\pi \mathrm{i} (k\cdot \xi ) \sigma _{-k,\alpha } \big ) \cdot \nabla \sigma _{k,\alpha } = (\sigma _{-k,\alpha }\cdot \nabla \xi ) \cdot \nabla \sigma _{k,\alpha }. \end{aligned}$$

Replacing \(\xi \) in (6.2) by \(\sigma _{-k,\alpha }\cdot \nabla \xi \) yields

$$\begin{aligned} I_2= 2\pi \mathrm{i} \big ((\sigma _{-k,\alpha }\cdot \nabla \xi ) \cdot k \big )\, \sigma _{k,\alpha }= 2\pi \mathrm{i} \big [k\cdot ( a_{k,\alpha }\cdot \nabla \xi ) \big ] a_{k,\alpha }. \end{aligned}$$

Summarizing the above computations we obtain the equality (6.4). \(\square \)

We want to find an a priori estimate for the solution to (6.1) with some heuristic computations below. By the Itô formula,

$$\begin{aligned} \begin{aligned} \mathrm{d}\Vert \xi \Vert _{L^2}^2&= -2\langle \xi , {\mathcal {L}}_u \xi \rangle _{L^2}\,\mathrm{d}t \\&\quad - 2(1+\nu ) \Vert \nabla \xi \Vert _{L^2}^2 \,\mathrm{d}t + \frac{ 2 C_\nu }{\Vert \theta \Vert _{\ell ^2}} \sum _{k, \alpha } \theta _k \big \langle \xi , {\mathcal {L}}_{\sigma _{k,\alpha }} \xi \big \rangle _{L^2} \, \mathrm{d}W^{k,\alpha }_t \\&\quad + \frac{2C_\nu ^2}{\Vert \theta \Vert _{\ell ^2}^2} \sum _{k, \alpha } \theta _k^2 \big \Vert {\mathcal {L}}_{\sigma _{k,\alpha }} \xi \big \Vert _{L^2}^2 \,\mathrm{d}t. \end{aligned} \end{aligned}$$
(6.6)

First, it is not difficult (cf. the proof of (3.5)) to show that

$$\begin{aligned} |\langle \xi , {\mathcal {L}}_u \xi \rangle _{L^2}| \le \frac{1}{2} \Vert \nabla \xi \Vert _{L^2}^{2} + C \Vert \xi \Vert _{L^2}^{6}. \end{aligned}$$
(6.7)

Next, we denote

$$\begin{aligned} \mathrm{d}M(t): = \frac{2C_\nu }{\Vert \theta \Vert _{\ell ^2}} \sum _{k, \alpha } \theta _k \big \langle \xi , {\mathcal {L}}_{\sigma _{k,\alpha }} \xi \big \rangle _{L^2} \, \mathrm{d}W^{k,\alpha }_t= - \frac{2C_\nu }{\Vert \theta \Vert _{\ell ^2}} \sum _{k,\alpha } \theta _k \big \langle \xi , \xi \cdot \nabla \sigma _{k,\alpha } \big \rangle _{L^2} \, \mathrm{d}W^{k,\alpha }_t \end{aligned}$$

the martingale part and

$$\begin{aligned} J(t)= \frac{2C_\nu ^2}{\Vert \theta \Vert _{\ell ^2}^2} \sum _{k,\alpha } \theta _k^2 \big \Vert {\mathcal {L}}_{\sigma _{k,\alpha }} \xi \big \Vert _{L^2}^2 =\frac{3\nu }{\Vert \theta \Vert _{\ell ^2}^2} \sum _{k,\alpha } \theta _k^2 \big \Vert {\mathcal {L}}_{\sigma _{k,\alpha }} \xi \big \Vert _{L^2}^2 . \end{aligned}$$

Then, since \(\langle \xi , \Delta \xi \rangle _{L^2}= - \Vert \nabla \xi \Vert _{L^2}^2\), we obtain from (6.6) and (6.7) that

$$\begin{aligned} \mathrm{d}\Vert \xi \Vert _{L^2}^2 \le -(1+2\nu ) \Vert \nabla \xi \Vert _{L^2}^2 \,\mathrm{d}t+ C \Vert \xi \Vert _{L^2}^{6}\,\mathrm{d}t + \mathrm{d}M(t) + J(t) \,\mathrm{d}t. \end{aligned}$$
(6.8)

Now we compute the term J(t).

Lemma 6.2

It holds that

$$\begin{aligned} J(t)= 2\nu \Vert \nabla \xi \Vert _{L^2}^2 + 4\nu \pi ^2 \frac{\Vert \theta \Vert _{h^1}^2}{\Vert \theta \Vert _{\ell ^2}^2} \Vert \xi \Vert _{L^2}^2, \end{aligned}$$

where

$$\begin{aligned} \Vert \theta \Vert _{h^1}^2 = \sum _{k\in {\mathbb {Z}}_0^3}\theta _k^2 |k|^2. \end{aligned}$$

Proof

We split J(t) as \(J(t)= \sum _{i=1}^3 J_i(t)\), where

$$\begin{aligned} \begin{aligned} J_1(t)&= \frac{3\nu }{\Vert \theta \Vert _{\ell ^2}^2} \sum _{k, \alpha } \theta _k^2 \big \Vert \sigma _{k,\alpha }\cdot \nabla \xi \big \Vert _{L^2}^2, \quad J_2(t)= \frac{3\nu }{\Vert \theta \Vert _{\ell ^2}^2} \sum _{k, \alpha } \theta _k^2 \big \Vert \xi \cdot \nabla \sigma _{k,\alpha } \big \Vert _{L^2}^2, \\ J_3(t)&= - \frac{3\nu }{\Vert \theta \Vert _{\ell ^2}^2} \sum _{k, \alpha } \theta _k^2 \big (\langle \sigma _{k,\alpha }\cdot \nabla \xi , \xi \cdot \nabla \sigma _{-k,\alpha } \rangle _{L^2} +\langle \sigma _{-k,\alpha }\cdot \nabla \xi , \xi \cdot \nabla \sigma _{k,\alpha } \rangle _{L^2} \big ). \end{aligned} \end{aligned}$$

Similarly as the proof of (2.3), we have

$$\begin{aligned} J_1(t)= 2\nu \Vert \nabla \xi \Vert _{L^2}^2. \end{aligned}$$

Next, by (6.2),

$$\begin{aligned} \big \Vert \xi \cdot \nabla \sigma _{k,\alpha } \big \Vert _{L^2}^2 = 4\pi ^2 \int _{{\mathbb {T}}^3} \big | (\xi \cdot k) \sigma _{k,\alpha } \big |^2\,\mathrm{d}x = 4\pi ^2 \int _{{\mathbb {T}}^3} (\xi \cdot k)^2 \,\mathrm{d}x. \end{aligned}$$

Thus,

$$\begin{aligned} \begin{aligned} J_2(t)&= \frac{3\nu }{\Vert \theta \Vert _{\ell ^2}^2} \sum _{k\in {\mathbb {Z}}^3_0} \theta _k^2 \times 4\pi ^2 \int _{{\mathbb {T}}^3} (\xi \cdot k)^2\,\mathrm{d}x = \frac{12\nu \pi ^2}{\Vert \theta \Vert _{\ell ^2}^2} \sum _{k\in {\mathbb {Z}}^3_0} \theta _k^2 \int _{{\mathbb {T}}^3} (\xi \cdot k)^2\,\mathrm{d}x. \end{aligned} \end{aligned}$$

Note that \((\xi \cdot k)^2 = \sum _{i, j=1}^3 k_i k_j \xi _i \xi _j\) and (cf. the computations below (2.3))

$$\begin{aligned} \sum _{k\in {\mathbb {Z}}^3_0} \theta _k^2 k_i k_j= {\left\{ \begin{array}{ll} 0, &{} i \ne j; \\ \frac{1}{3} \sum _{k\in {\mathbb {Z}}^3_0} \theta _k^2 |k|^2= \frac{1}{3} \Vert \theta \Vert _{h^1}^2, &{} i= j. \end{array}\right. } \end{aligned}$$

Therefore,

$$\begin{aligned} J_2(t)= \frac{12\nu \pi ^2}{\Vert \theta \Vert _{\ell ^2}^2} \sum _{i=1}^3 \frac{1}{3} \Vert \theta \Vert _{h^1}^2 \int _{{\mathbb {T}}^3} \xi _i^2\,\mathrm{d}x= 4\nu \pi ^2 \frac{\Vert \theta \Vert _{h^1}^2}{\Vert \theta \Vert _{\ell ^2}^2} \Vert \xi \Vert _{L^2}^2. \end{aligned}$$

Finally, by (6.2) and the definition of the vector fields \(\sigma _{k,\alpha }\), we have

$$\begin{aligned} \langle \sigma _{k,\alpha }\cdot \nabla \xi , \xi \cdot \nabla \sigma _{-k,\alpha } \rangle _{L^2} = -2\pi \mathrm{i} \int _{{\mathbb {T}}^3} (\xi \cdot k) (a_{k,\alpha }\cdot \nabla \xi ) \cdot a_{k,\alpha } \,\mathrm{d}x. \end{aligned}$$

In the same way,

$$\begin{aligned} \langle \sigma _{-k,\alpha }\cdot \nabla \xi , \xi \cdot \nabla \sigma _{k,\alpha } \rangle _{L^2}= 2\pi \mathrm{i} \int _{{\mathbb {T}}^3} (\xi \cdot k) (a_{k,\alpha }\cdot \nabla \xi ) \cdot a_{k,\alpha } \,\mathrm{d}x. \end{aligned}$$

Hence \(J_3(t)\) vanishes. Summarizing these arguments we complete the proof. \(\square \)

Therefore, the inequality (6.8) reduces to

$$\begin{aligned} \mathrm{d}\Vert \xi \Vert _{L^2}^2 \le - \Vert \nabla \xi \Vert _{L^2}^2 \,\mathrm{d}t+ C \Vert \xi \Vert _{L^2}^{6}\,\mathrm{d}t + \mathrm{d}M(t) + 4\nu \pi ^2 \frac{\Vert \theta \Vert _{h^1}^2}{\Vert \theta \Vert _{\ell ^2}^2} \Vert \xi \Vert _{L^2}^2 \,\mathrm{d}t. \end{aligned}$$

The ratio \(\frac{\Vert \theta \Vert _{h^1}^2}{\Vert \theta \Vert _{\ell ^2}^2}\) spoils the a priori estimate, since the sequence \(\{\theta ^N \}_{N\ge 1}\) we take in our limit process has always the property

$$\begin{aligned} \lim _{N\rightarrow \infty } \frac{\Vert \theta ^N \Vert _{h^1}^2}{\Vert \theta ^N \Vert _{\ell ^2}^2} =\infty . \end{aligned}$$

Appendix 3: An incomplete attempt to motivate transport noise

We advise the reader that the argument given in this section is a sort of cartoon based on imagination, and a potentially rigorous scaling limit behind it would be presumably much more intricate than what is explained, or maybe even impossible.

A fact, rigorous in several function spaces, is that given two vector fields AB in \({\mathbb {R}}^{3}\), the condition

$$\begin{aligned} {\mathcal {L}}_{A}B=\Pi \left( A\cdot \nabla B\right) \end{aligned}$$
(7.1)

is equivalent to

$$\begin{aligned} B\cdot \nabla A=\nabla q \end{aligned}$$

for some scalar function q; the particular case when \(\nabla q=0\) is implied by a “2D structure”

$$\begin{aligned} B\left( x\right) =b\left( x\right) e,\quad A\left( x\right) =A\left( \pi _{e^{\perp }}x\right) \end{aligned}$$
(7.2)

where e is a given unitary vector, \(b\left( x\right) \) is a scalar function on \({\mathbb {R}}^{3}\) (hence the vector field B points always in the direction e) and the improper notation \(A\left( x\right) =A\left( \pi _{e^{\perp }}x\right) \) means that A depends only on the projection of x on the plane orthogonal to e (namely A is independent of the coordinate along e; this implies that the directional derivative of A in the direction e is zero, which is precisely \(B\cdot \nabla A=0\)). What we describe below is a sort of local 2D structure, with different orientations e at different points, in which the identity (7.1) could be approximately satisfied.

Assume to observe a fluid where the vorticity field \(\xi \) is made of two components

$$\begin{aligned} \xi =\xi _{L}+\xi _{S} \end{aligned}$$

where the large-scale component \(\xi _{L}\) is the sum of slowly varying smoothed vortex filaments \(\xi _{L}^{i}\)

$$\begin{aligned} \xi _{L}=\sum _{i}\xi _{L}^{i} \end{aligned}$$

and the small-scale component \(\xi _{S}\) is a fast-varying field. By smoothed vortex filament we mean a vortex structure strongly concentrated along a vortex line; in the spirit of this cartoon we do not give any precise definition, but vortex filaments, although extremely difficult to define and describe, are commonly observed structures in complex fluids (see [42]). We need to qualify the filaments as smoothed because viscosity does not allow for idealized filaments concentrated over lines. Corresponding to the vorticity fields there are velocity fields obtained by Biot–Savart law, \(u=u_{L}+u_{S}\).

Consider a point \(x_{0}\) close to the core of a smoothed vortex filament \(\xi _{L}^{i}\), consider a neighbourhood \({\mathcal {U}}\left( x_{0}\right) \) of \(x_{0}\) and imagine a blow-up, a scaling such that we observe \({\mathcal {U}} \left( x_{0}\right) \) as if it were the full space. If the vortex filaments are sufficiently thin, separated, regular and slowly moving compared to the fast component \(u_{S}\), in \({\mathcal {U}}\left( x_{0}\right) \) (which now looks as the entire space) the vorticity is very close to zero everywhere except along the line spanned by the vector \(e=\xi _{L}^{i}\left( x_{0}\right) \); moreover, we may think to consider the full system on a time scale where the large-scale objects \({\mathcal {U}}\left( x_{0}\right) \), \(\xi _{L}^{i}\left( x_{0}\right) \) etc. do not change while the small-scale objects \(\xi _{S},u_{S}\) change. The local picture of the small-scale fluid \(u_{S}\) in \({\mathcal {U}}\left( x_{0}\right) \) is thus of a 3D fluid subject to a constant strong rotation around the vector e. If such a fluid, namely \(u_{S}|_{{\mathcal {U}}\left( x_{0}\right) }\), would be isolated from any other input and interaction, it would become approximatively averaged in the direction e, like the field A in (7.2). This has been rigorously proved in several works, see for instance [3] (see also [22] in a stochastic framework). Obviously we do not mean that the global field \(u_{S}\) is almost two-dimensional: only at local level it has a tendency to average in the direction of \(\xi _{L}^{i}\left( x_{0}\right) \); this vector changes orientation from a small region to another. When this happens, we have \(\xi _{L}^{i}\left( x_{0}\right) \cdot \nabla u_{S}\left( x_{0}\right) \sim 0\). We have argued in the proximity of a vortex core; far from filaments \(\xi _{L}^{i}\left( x_{0}\right) \cdot \nabla u_{S}\left( x_{0}\right) \) is small just because \(\xi _{L}\) is almost zero by itself. We deduce that everywhere

$$\begin{aligned} \xi _{L}\left( x\right) \cdot \nabla u_{S}\left( x\right) \sim 0. \end{aligned}$$
(7.3)

We ignore whether it is possible to establish a more rigorous derivation of such a fact by a proper scaling limit and maybe an argument similar to the concept of local equilibrium in the statistical mechanics of particle systems, where the local convergence to equilibrium is replaced by the “vertical averaging” property described above.

Let us derive a consequence of (7.3). Given a decomposition

$$\begin{aligned} \xi \left( 0\right) =\xi _{L}\left( 0\right) +\xi _{S}\left( 0\right) \end{aligned}$$

of an initial condition \(\xi \left( 0\right) \), if the system

$$\begin{aligned} \partial _{t}\xi _{L}+{\mathcal {L}}_{u_{L}}\xi _{L}+{\mathcal {L}}_{u_{S}}\xi _{L}&=\Delta \xi _{L}\\ \partial _{t}\xi _{S}+{\mathcal {L}}_{u_{S}}\xi _{S}+{\mathcal {L}}_{u_{L}}\xi _{S}&=\Delta \xi _{S} \end{aligned}$$

with initial condition \(\left( \xi _{L}\left( 0\right) ,\xi _{S}\left( 0\right) \right) \) has a solution, then \(\xi =\xi _{L}+\xi _{S}\) is a solution of the full 3D Navier–Stokes equations, solution decomposed in the two “scales” \(\xi _{L}\) and \(\xi _{S}\). Consider the first equation, for the large scales. We have

$$\begin{aligned} {\mathcal {L}}_{u_{S}}\xi _{L}=u_{S}\cdot \nabla \xi _{L}-\xi _{L}\cdot \nabla u_{S}. \end{aligned}$$

We may also write

$$\begin{aligned} {\mathcal {L}}_{u_{S}}\xi _{L}=\Pi \left( u_{S}\cdot \nabla \xi _{L}\right) -\Pi \left( \xi _{L}\cdot \nabla u_{S}\right) \end{aligned}$$

since \(\Pi \left( {\mathcal {L}}_{u_{S}}\xi _{L}\right) ={\mathcal {L}}_{u_{S}} \xi _{L}\) (but this is not true separately for the two addends). The equation for the large scales then is

$$\begin{aligned} \partial _{t}\xi _{L}+{\mathcal {L}}_{u_{L}}\xi _{L}+\Pi \left( u_{S}\cdot \nabla \xi _{L}\right) =\Delta \xi _{L}+\Pi \left( \xi _{L}\cdot \nabla u_{S}\right) . \end{aligned}$$

Assume we may apply the arguments described above. We get (approximately) the equation

$$\begin{aligned} \partial _{t}\xi _{L}+{\mathcal {L}}_{u_{L}}\xi _{L}+\Pi \left( u_{S}\cdot \nabla \xi _{L}\right) =\Delta \xi _{L}. \end{aligned}$$

The model considered in this work corresponds to the idealization when \(u_{S}\) is replaced by a white noise in time, idealization reminiscent of stochastic reduction techniques like those more carefully developed in [35]. To be fair, let us notice that the isotropic noise considered in our work is incompatible with the orthogonality conditions (7.3), making the above justification still incomplete even at a very heuristic ground.

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Flandoli, F., Luo, D. High mode transport noise improves vorticity blow-up control in 3D Navier–Stokes equations. Probab. Theory Relat. Fields 180, 309–363 (2021). https://doi.org/10.1007/s00440-021-01037-5

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Keywords

  • 3D Navier–Stokes equations
  • Well posedness
  • Regularization by noise
  • Transport noise
  • Vorticity blow-up control

Mathematics Subject Classification

  • Primary 60H15
  • Secondary 76D05