Diffusion and mixing in fluid flow via the resolvent estimate

Abstract

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.

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References

  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

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Acknowledgements

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

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Correspondence to Dongyi Wei.

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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

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Keywords

  • mixing
  • resolvent estimate
  • shear flows
  • Weierstrass functions

MSC(2010)

  • 35Q35