Skip to main content
SpringerLink
Log in

Rate of the Enhanced Dissipation for the Two-jet Kolmogorov Type Flow on the Unit Sphere

  • Published:
Journal of Mathematical Fluid Mechanics Aims and scope Submit manuscript

Abstract

We study the enhanced dissipation for the two-jet Kolmogorov type flow which is a stationary solution to the Navier–Stokes equations on the two-dimensional unit sphere given by the zonal spherical harmonic function of degree two. Based on the pseudospectral bound method developed by Ibrahim, Maekawa, and Masmoudi [14] and a modified version of the Gearhart–Prüss type theorem shown by Wei [50], we derive an estimate for the resolvent of the linearized operator along the imaginary axis and show that a solution to the linearized equation rapidly decays at the rate \(O(e^{-c\sqrt{\nu }\,t})\) when the viscosity coefficient \(\nu \) is sufficiently small as in the case of the plane Kolmogorov flow.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aris, R.: Vectors, tensors and the basic equations of fluid mechanics. Mineola, NY: Dover Publications, reprint of the 1962 original edition, (1989)

  2. Aubin, T.: Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics., Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  3. 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 143(5), 905–927 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cao, C., Rammaha, M.A., Titi, E.S.: The Navier-Stokes equations on the rotating \(2\)-D sphere: Gevrey regularity and asymptotic degrees of freedom. Z. Angew. Math. Phys. 50(3), 341–360 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chan, C.H., Czubak, M.: Non-uniqueness of the Leray-Hopf solutions in the hyperbolic setting. Dyn. Partial Differ. Equ. 10(1), 43–77 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chan, C.H., Yoneda, T.: On the stationary Navier-Stokes flow with isotropic streamlines in all latitudes on a sphere or a 2D hyperbolic space. Dyn. Partial Differ. Equ. 10(3), 209–254 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chan, C.H., Czubak, M., Disconzi, M.M.: The formulation of the Navier-Stokes equations on Riemannian manifolds. J. Geom. Phys. 121, 335–346 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Dindoš, M., Mitrea, M.: The stationary Navier-Stokes system in nonsmooth manifolds: the Poisson problem in Lipschitz and \(C^1\) domains. Arch. Ration. Mech. Anal. 174(1), 1–47 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Duduchava, L.R., Mitrea, D., Mitrea, M.: Differential operators and boundary value problems on hypersurfaces. Math. Nachr. 279(9–10), 996–1023 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ebin, D.G., Marsden, J.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. of Math. 2(92), 102–163 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  12. Engel, K.-J., Nagel, R.: One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt

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

    MathSciNet  MATH  Google Scholar 

  14. Ibrahim, S., Maekawa, Y., Masmoudi, N.: On pseudospectral bound for non-selfadjoint operators and its application to stability of Kolmogorov flows. Ann. PDE, 5(2):Paper No. 14, 84, (2019)

  15. Il’in, A.A.: Navier-Stokes and Euler equations on two-dimensional closed manifolds. Mat. Sb. 181(4), 521–539 (1990)

    MATH  Google Scholar 

  16. Il’n, A.A., Filatov, A.N.: Unique solvability of the Navier-Stokes equations on a two-dimensional sphere. Dokl. Akad. Nauk SSSR 301(1), 18–22 (1988)

    ADS  MathSciNet  Google Scholar 

  17. Ilyin, A.A.: Stability and instability of generalized Kolmogorov flows on the two-dimensional sphere. Adv. Differ. Equ. 9(9–10), 979–1008 (2004)

    MathSciNet  MATH  Google Scholar 

  18. Iudovich, V.I.: Example of the generation of a secondary stationary or periodic flow when there is loss of stability of the laminar flow of a viscous incompressible fluid. J. Appl. Math. Mech. 29(3), 527–544 (1965)

    Article  Google Scholar 

  19. Kato, T.: Perturbation theory for linear operators. Springer-Verlag, Berlin-New York, second edition, (1976). Grundlehren der Mathematischen Wissenschaften, Band 132

  20. Khesin, B., Misiołek, G.: Euler and Navier-Stokes equations on the hyperbolic plane. Proc. Natl. Acad. Sci. USA 109(45), 18324–18326 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  21. Kohr, M., Wendland, W.L.: Variational approach for the Stokes and Navier-Stokes systems with nonsmooth coefficients in Lipschitz domains on compact Riemannian manifolds. Calc. Var. Partial Differ. Equ. 57(6), 41 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lebedev, N.N.: Special functions and their applications. Dover Publications Inc, New York, Revised edition, translated from the Russian and edited by Richard A. Unabridged and corrected republication, Silverman (1972)

  23. Lee, J.M.: Introduction to smooth manifolds. Graduate Texts in Mathematics, vol. 218, 2nd edn. Springer, New York (2013)

    MATH  Google Scholar 

  24. Lee, J.M.: Introduction to Riemannian manifolds. Graduate Texts in Mathematics, vol. 176, 2nd edn. Springer, Cham (2018)

    Book  MATH  Google Scholar 

  25. Li, T., Wei, D., Zhang, Z.: Pseudospectral bound and transition threshold for the \(3\)D Kolmogorov flow. Comm. Pure Appl. Math. 73, 465–557 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lichtenfelz, L.A.: Nonuniqueness of solutions of the Navier-Stokes equations on Riemannian manifolds. Ann. Global Anal. Geom. 50(3), 237–248 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lin, Z., Xu, M.: Metastability of Kolmogorov flows and inviscid damping of shear flows. Arch. Ration. Mech. Anal. 231(3), 1811–1852 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  28. Marchioro, C.: An example of absence of turbulence for any Reynolds number. Comm. Math. Phys. 105(1), 99–106 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Matsuda, M., Miyatake, S.: Bifurcation analysis of Kolmogorov flows. Tohoku Math. J. (2) 54(3), 329–365 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mešalkin, L.D., Sinaĭ, J.G.: Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous liquid. J. Appl. Math. Mech. 25, 1700–1705 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mitrea, M., Taylor, M.: Navier-Stokes equations on Lipschitz domains in Riemannian manifolds. Math. Ann. 321(4), 955–987 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. Miura, T.-H.: Linear stability and enhanced dissipation for the two-jet kolmogorov type flow on the unit sphere. J. Funct. Anal. to appear

  33. Nagasawa, T.: Construction of weak solutions of the Navier-Stokes equations on Riemannian manifold by minimizing variational functionals. Adv. Math. Sci. Appl. 9(1), 51–71 (1999)

    MathSciNet  MATH  Google Scholar 

  34. NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.1.1 of 2021-03-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds

  35. Okamoto, H., Shōji, M.: Bifurcation diagrams in Kolmogorov’s problem of viscous incompressible fluid on \(2\)-D flat tori. Japan J. Indust. Appl. Math. 10(2), 191–218 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  36. Pierfelice, V.: The incompressible Navier-Stokes equations on non-compact manifolds. J. Geom. Anal. 27(1), 577–617 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  37. Priebe, V.: Solvability of the Navier-Stokes equations on manifolds with boundary. Manuscripta Math. 83(2), 145–159 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  38. Prüss, J., Simonett, G., Wilke, M.: On the Navier-Stokes equations on surfaces. J. Evol. Equ. 21(3), 3153–3179 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  39. Samavaki, M., Tuomela, J.: Navier-Stokes equations on Riemannian manifolds. J. Geom. Phys. 148, 103543 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  40. Sasaki, E., Takehiro, S., Yamada, M.: Linear stability of viscous zonal jet flows on a rotating sphere. J. Phys. Soc. Japan 82(9), 094402 (2013)

    Article  ADS  Google Scholar 

  41. Sasaki, E., Takehiro, S., Yamada, M.: Bifurcation structure of two-dimensional viscous zonal flows on a rotating sphere. J. Fluid Mech. 774, 224–244 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Skiba, Y.N.: Mathematical problems of the dynamics of incompressible fluid on a rotating sphere. Springer, Cham (2017)

    Book  MATH  Google Scholar 

  43. Taylor, M.E.: Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations. Comm. Partial Differ. Equ. 17(9–10), 1407–1456 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  44. Taylor, M.E.: Partial differential equations III. Nonlinear equations. Applied Mathematical Sciences, vol. 117, 2nd edn. Springer, New York (2011)

    Book  MATH  Google Scholar 

  45. Taylor, M.: Euler equation on a rotating surface. J. Funct. Anal. 270(10), 3884–3945 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  46. Temam, R., Wang, S.H.: Inertial forms of Navier-Stokes equations on the sphere. J. Funct. Anal. 117(1), 215–242 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  47. Teschl, G.: Mathematical methods in quantum mechanics, volume 157 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, (2014). With applications to Schrödinger operators

  48. Varshalovich, D. A., Moskalev, A. N., Khersonskiĭ, V. K.: Quantum theory of angular momentum. World Scientific Publishing Co., Inc., Teaneck, NJ, (1988). Irreducible tensors, spherical harmonics, vector coupling coefficients, \(3nj\) symbols, Translated from the Russian

  49. Villani, C.: Hypocoercivity. Mem. Amer. Math. Soc., 202(950):iv+141, (2009)

  50. Wei, D.: Diffusion and mixing in fluid flow via the resolvent estimate. Sci. China Math. 64(3), 507–518 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  51. Wei, D., Zhang, Z.: Enhanced dissipation for the Kolmogorov flow via the hypocoercivity method. Sci. China Math. 62(6), 1219–1232 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  52. Wei, D., Zhang, Z., Zhao, W.: Linear inviscid damping and enhanced dissipation for the Kolmogorov flow. Adv. Math. 362, 106963 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  53. Wirosoetisno, D.: Navier-Stokes equations on a rapidly rotating sphere. Discrete Contin. Dyn. Syst. Ser. B 20(4), 1251–1259 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  54. Zlatoš, A.: Diffusion in fluid flow: dissipation enhancement by flows in 2D. Comm. Partial Differ. Equ. 35(3), 496–534 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The work of the first author was supported by JSPS KAKENHI Grant Numbers 20K03698, 19H05597, 20H00118. Also, the work of the second author was supported by Grant-in-Aid for JSPS Fellows No. 19J00693. The authors would like to thank an anonymous referee for a careful reading and valuable comments on this work.

Author information

Author notes

  1. Tatsu-Hiko Miura

    Present address: Graduate School of Science and Technology, Hirosaki University, 3, Bunkyo-cho, Hirosaki-shi, Aomori, 036-8561, Japan

Authors and Affiliations

  1. Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan

    Yasunori Maekawa & Tatsu-Hiko Miura

Authors
  1. Yasunori Maekawa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tatsu-Hiko Miura
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yasunori Maekawa.

Ethics declarations

Conflicts of interest

There is no conflict of interest. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Additional information

Communicated by T. Ozawa.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the topical collection “Yoshihiro Shibata” edited by Tohru Ozawa.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Maekawa, Y., Miura, TH. Rate of the Enhanced Dissipation for the Two-jet Kolmogorov Type Flow on the Unit Sphere. J. Math. Fluid Mech. 24, 92 (2022). https://doi.org/10.1007/s00021-022-00718-y

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00021-022-00718-y

Keywords

Mathematics Subject Classification

Search

Navigation