Diffusion and mixing in fluid flow via the resolvent estimate


In this paper, we first present a Gearhart-Prüss type theorem with a sharp bound for m-accretive operators. Then we give two applications: (1) we give a simple proof of the result proved by Constantin et al. on relaxation enhancement induced by incompressible flows; (2) we show that shear flows with a class of Weierstrass functions obey logarithmically fast dissipation time-scales.

This is a preview of subscription content, access via your institution.


  1. 1

    Beck M, Wayne C E. Metastability and rapid convergence to quasi-stationary bar states for the two-dimensional Navier-Stokes equations. Proc Roy Soc Edinburgh Sect A, 2013, 143: 905–927

    MathSciNet  Article  Google Scholar 

  2. 2

    Bedrossian J, Coti Zelati M. Enhanced dissipation, hypoellipticity, and anomalous small noise inviscid limits in shear flows. Arch Ration Mech Anal, 2017, 224: 1161–1204

    MathSciNet  Article  Google Scholar 

  3. 3

    Bedrossian J, Germain P, Masmoudi N. On the stability threshold for the 3D Couette flow in Sobolev regularity. Ann of Math (2), 2017, 185: 541–608

    MathSciNet  Article  Google Scholar 

  4. 4

    Bedrossian J, He S. Suppression of blow-up in Patlak-Keller-Segel via shear flows. SIAM J Math Anal, 2017, 49: 4722–4766

    MathSciNet  Article  Google Scholar 

  5. 5

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

    MathSciNet  Article  Google Scholar 

  6. 6

    Coti Zelati M, Delgadino M G, Elgindi T M. On the relation between enhanced dissipation time-scales and mixing rates. ArXiv:1806.03258, 2018

  7. 7

    Engel K J, Nagel R. One-parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, 194. New York: Springer-Verlag, 2000

    Google Scholar 

  8. 8

    Gallagher I, Gallay T, Nier F. Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator. Int Math Res Not IMRN, 2009, 12: 2147–2199

    MathSciNet  MATH  Google Scholar 

  9. 9

    Grenier E, Nguyen T, Rousset F, et al. Linear inviscid damping and enhanced viscous dissipation of shear flows by using the conjugate operator method. ArXiv:1804.08291, 2018

  10. 10

    Helffer B, Sjöstrand J. From resolvent bounds to semigroup bounds. ArXiv:1001.4171, 2010

  11. 11

    Ibrahim S, Maekawa Y, Masmoudi N. On pseudospectral bound for non-selfadjoint operators and its application to stability of Kolmogorov flows. ArXiv:1710.05132, 2017

  12. 12

    Kato T. Perturbation Theory for Linear Operators. Grundlehren der mathematischen Wissenschaften, vol. 132. Berlin: Springer, 1966

    Google Scholar 

  13. 13

    Kiselev A, Zlatoš A. Quenching of combustion by shear flows. Duke Math J, 2006, 132: 49–72

    MathSciNet  Article  Google Scholar 

  14. 14

    Li T, Wei D, Zhang Z. Pseudospectral bound and transition threshold for the 3D Kolmogorov flow. ArXiv:1801.05645, 2018

  15. 15

    Villani C. Hypocoercivity. Memoirs of the American Mathematical Society, vol. 202. Providence: Amer Math Soc, 2009

    Google Scholar 

  16. 16

    Wei D, Zhang Z. Transition threshold for the 3D Couette flow in Sobolev space. ArXiv:1803.01359, 2018

  17. 17

    Wei D, Zhang Z, Zhao W. Linear inviscid damping and enhanced dissipation for the Kolmogorov flow. ArXiv:1711.01822, 2017

Download references


This work was supported by China Postdoctoral Science Foundation (Grant No. 2018M 630016). The author thanks Professor Zhifei Zhang for many helpful suggestions.

Author information



Corresponding author

Correspondence to Dongyi Wei.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wei, D. Diffusion and mixing in fluid flow via the resolvent estimate. Sci. China Math. 64, 507–518 (2021). https://doi.org/10.1007/s11425-018-9461-8

Download citation


  • mixing
  • resolvent estimate
  • shear flows
  • Weierstrass functions


  • 35Q35