On the Sobolev Stability Threshold of 3D Couette Flow in a Uniform Magnetic Field

Abstract

We study the stability of the Couette flow \((y,0,0)^T\) in the presence of a uniform magnetic field \(\alpha (\sigma , 0, 1)\) on \({{\mathbb {T}}}\times {{\mathbb {R}}}\times {{\mathbb {T}}}\) using the 3D incompressible magnetohydrodynamics (MHD) equations. We consider the inviscid, ideal conductor limit \(\mathbf{Re} ^{-1}\), \(\mathbf{R }_m^{-1} \ll 1\) and prove that for strong and suitably oriented background fields the Couette flow is asymptotically stable to perturbations small in the Sobolev space \(H^N\). More precisely, we show that if \(\mathbf{Re} ^{-1} = \mathbf{R }_m^{-1} \in (0,1]\), \(\alpha > 0\) and \(N > 0\) are sufficiently large, \(\sigma \in {{\mathbb {R}}}{\setminus } {\mathbb {Q}}\) satisfies a generic Diophantine condition, and the initial perturbations \(u_{\text{ in }}\) and \(b_{\text{ in }}\) to the Couette flow and magnetic field, respectively, satisfy \(\Vert u_{\text{ in }}\Vert _{H^N} + \Vert b_{\text{ in }}\Vert _{H^N} = \epsilon \ll \mathbf{Re} ^{-1}\), then the resulting solution to the 3D MHD equations is global in time and the perturbations \(u(t,x+yt,y,z)\) and \(b(t,x+yt,y,z)\) remain \({\mathcal {O}}(\mathbf{Re} ^{-1})\) in \(H^{N'}\) for some \(1 \ll N'(\sigma ) < N\). Our proof establishes enhanced dissipation estimates describing the decay of the x-dependent modes on the timescale \(t \sim \mathbf{Re} ^{1/3}\), as well as inviscid damping of the velocity and magnetic field with a rate that agrees with the prediction of the linear stability analysis. In the Navier–Stokes case, high regularity control on the perturbation in a coordinate system adapted to the mixing of the Couette flow is known only under the stronger assumption \(\epsilon \ll \mathbf{Re} ^{-3/2}\) (Bedrossian et al. in Ann. Math. 185(2): 541–608, 2017). The improvement in the MHD setting is possible because the magnetic field induces time oscillations that partially suppress the lift-up effect, which is the primary transient growth mechanism for the Navier–Stokes equations linearized around Couette flow.

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

References

  1. 1.

    Alinhac, S.: The null condition for quasilinear wave equations in two space dimensions i. Invent. Math. 145(3), 597–618 (2001)

    ADS  MathSciNet  Article  Google Scholar 

  2. 2.

    Bedrossian, J., Germain, P., Masmoudi, N.: Dynamics near the subcritical transition of the 3D Couette flow I: below threshold. Mem. Amer. Math. Soc. (to appear) (2015). arXiv:1506.03720

  3. 3.

    Bedrossian, J., Germain, P., Masmoudi, N.: Dynamics near the subcritical transition of the 3D Couette flow II: above threshold (2015). arXiv:1506.03721

  4. 4.

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

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bedrossian, J., Germain, P., Masmoudi, N.: Stability of the Couette flow at high Reynolds numbers in two dimensions and three dimensions. Bull. Am. Math. Soc. 56, 373–414 (2019)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bedrossian, J., Masmoudi, N.: Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations. Publications Mathématiques de l’IHÉS 122(1), 195–300 (2015)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Bedrossian, J., Masmoudi, N., Vicol, V.: Enhanced dissipation and inviscid damping in the inviscid limit of the Navier–Stokes equations near the 2D Couette flow. Arch. Ration. Mech. Anal. 216(3), 1087–1159 (2016)

    Article  Google Scholar 

  8. 8.

    Bedrossian, J., Vicol, V., Wang, F.: The Sobolev stability threshold for 2D shear flows near Couette. J. Nonlinear Sci. 28, 2051–2075 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  9. 9.

    Carlson, D.R., Widnall, S.E., Peeters, M.F.: Aflow-visualization study of transition in plane poiseuille flow. J. Fluid Mech. 121, 487–505 (1982)

    ADS  Article  Google Scholar 

  10. 10.

    Cassels, J.: An Introduction to Diophantine Approximation. Cambridge Tracts in Mathematics and Mathematical Physics. Cambridge University Press, Cambridge (1957)

    Google Scholar 

  11. 11.

    Chandrasekhar, S.: The stability of viscous flow between rotating cylinders in the presence of a magnetic field. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 216(1126), 293–309 (1953)

    ADS  MathSciNet  MATH  Google Scholar 

  12. 12.

    Chandrasekhar, S.: The stability of non-dissipative couette flow in hydromagnetics. Proc. Natl. Acad. Sci. 46(2), 253–257 (1960)

    ADS  MathSciNet  Article  Google Scholar 

  13. 13.

    Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Dover Books on Physics Series. Dover Publications, New York (1981)

    Google Scholar 

  14. 14.

    Davidson, P.A.: An Introduction to Magnetohydrodynamics. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  15. 15.

    Deng, Y., Masmoudi, N.: Long time instability of Couette flow in low Gevrey spaces (2018). arXiv:1803.01246

  16. 16.

    Drazin, P.G., Reid, W.H.: Hydrodynamic Stability. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, Cambridge (1981)

    Google Scholar 

  17. 17.

    Ellingsen, T., Palm, E.: Stability of linear flow. Phys. Fluids 18(4), 487–488 (1975)

    ADS  Article  Google Scholar 

  18. 18.

    Germain, P.: Global existence for coupled Klein–Gordon equations with different speeds. Annales de l’Institut Fourier 61(6), 2463–2506 (2011)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Germain, P.: Space-time resonances (2011). arXiv:1102.1695

  20. 20.

    Germain, P., Masmoudi, N.: Global existence for the Euler–Maxwell system. Annales Scientifiques de l’Ecole Normale Superieure 47(3), 469–503 (2014)

    MathSciNet  Article  Google Scholar 

  21. 21.

    He, L.-B., Xu, L., Yu, P.: On global dynamics of three dimensional magnetohydrodynamics: nonlinear stability of alfvén waves. Ann. PDE 4(1), 5 (2017)

    Article  Google Scholar 

  22. 22.

    Hughes, D., Tobias, S.: On the instability of magnetohydrodynamic shear flows. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 457(2010), 1365–1384 (2001)

    ADS  MathSciNet  Article  Google Scholar 

  23. 23.

    Hunt, J.: On the stability of parallel flows with parallel magnetic fields. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 293(1434), 342–358 (1966)

    ADS  Google Scholar 

  24. 24.

    Ionescu, A., Jia, H.: Inviscid damping near shear flows in a channel (2018). arXiv:1808.04026

  25. 25.

    Kelvin, L.: Stability of fluid motion-rectilinear motion of viscous fluid between two parallel plates. Philos. Mag. 24, 188 (1887)

    Article  Google Scholar 

  26. 26.

    Liu, Y., Chen, Z.H., Zhang, H.H., Lin, Z.Y.: Physical effects of magnetic fields on the Kelvin–Helmholtz instability in a free shear layer. Phys. Fluids 30(4), 044102 (2018)

    ADS  Article  Google Scholar 

  27. 27.

    Majda, A., Bertozzi, A.: Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  28. 28.

    Mamatsashvili, G.R., Gogichaishvili, D.Z., Chagelishvili, G.D., Horton, W.: Nonlinear transverse cascade and two-dimensional magnetohydrodynamic subcritical turbulence in plane shear flows. Phys. Rev. E 89, 043101 (2014)

    ADS  Article  Google Scholar 

  29. 29.

    Orr, W.: The stability or instability of steady motions of a perfect liquid and of a viscous liquid, Part I: a perfect liquid. Proc. R. Ir. Acad. Sec. A Math. Phys. Sci. 27, 9–68 (1907)

    Google Scholar 

  30. 30.

    Stuart, J.: On the stability of viscous flow between parallel planes in the presence of a co-planar magnetic field. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 221(1145), 189–206 (1954)

    ADS  MathSciNet  MATH  Google Scholar 

  31. 31.

    Trefethen, L.N., Trefethen, A.E., Reddy, S.C., Driscoll, T.A.: Hydrodynamic stability without eigenvalues. Science 261(5121), 578–584 (1993)

    ADS  MathSciNet  Article  Google Scholar 

  32. 32.

    Velikhov, E.: Stability of an ideally conducting liquid flowing between rotating cylinders in a magnetic field. Zhur. Eksptl’. i Teoret. Fiz. 36, 05 (1959)

    Google Scholar 

  33. 33.

    Wei, D., Zhang, Z.: Global well-posedness of the MHD equations in a homogeneous magnetic field. Anal. PDE 10(6), 1361–1406 (2017)

    MathSciNet  Article  Google Scholar 

  34. 34.

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

  35. 35.

    Yaglom, A., Frisch, U.: Hydrodynamic Instability and Transition to Turbulence. Fluid Mechanics and Its Applications. Springer, Dordrecht (2012)

    Google Scholar 

Download references

Acknowledgements

The author would like to thank his advisor, Jacob Bedrossian, for suggesting an MHD stability problem and providing guidance throughout. This work was partially supported by Jacob Bedrossian’s NSF CAREER Grant DMS-1552826 and NSF RNMS #1107444 (Ki-Net).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Kyle Liss.

Additional information

Publisher's Note

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

Communicated by C. De Lellis

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Liss, K. On the Sobolev Stability Threshold of 3D Couette Flow in a Uniform Magnetic Field . Commun. Math. Phys. 377, 859–908 (2020). https://doi.org/10.1007/s00220-020-03768-3

Download citation