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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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
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
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
Bedrossian J, He S. Suppression of blow-up in Patlak-Keller-Segel via shear flows. SIAM J Math Anal, 2017, 49: 4722–4766
Constantin P, Kiselev A, Ryzhik L, et al. Diffusion and mixing in fluid flow. Ann of Math (2), 2008, 168: 643–674
Coti Zelati M, Delgadino M G, Elgindi T M. On the relation between enhanced dissipation time-scales and mixing rates. ArXiv:1806.03258, 2018
Engel K J, Nagel R. One-parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, 194. New York: Springer-Verlag, 2000
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
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
Helffer B, Sjöstrand J. From resolvent bounds to semigroup bounds. ArXiv:1001.4171, 2010
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
Kato T. Perturbation Theory for Linear Operators. Grundlehren der mathematischen Wissenschaften, vol. 132. Berlin: Springer, 1966
Kiselev A, Zlatoš A. Quenching of combustion by shear flows. Duke Math J, 2006, 132: 49–72
Li T, Wei D, Zhang Z. Pseudospectral bound and transition threshold for the 3D Kolmogorov flow. ArXiv:1801.05645, 2018
Villani C. Hypocoercivity. Memoirs of the American Mathematical Society, vol. 202. Providence: Amer Math Soc, 2009
Wei D, Zhang Z. Transition threshold for the 3D Couette flow in Sobolev space. ArXiv:1803.01359, 2018
Wei D, Zhang Z, Zhao W. Linear inviscid damping and enhanced dissipation for the Kolmogorov flow. ArXiv:1711.01822, 2017
Acknowledgements
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
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-018-9461-8