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On the Forced Euler and Navier–Stokes Equations: Linear Damping and Modified Scattering

  • Christian ZillingerEmail author
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Abstract

We study the asymptotic behavior of the forced linear Euler and nonlinear Navier–Stokes equations close to Couette flow on \(\mathbb {T}\times I\). As our main result we show that for smooth time-periodic forcing linear inviscid damping persists, i.e. the velocity field (weakly) asymptotically converges. However, stability and scattering to the transport problem fail in \(H^{s}, s>-1\). We further show that this behavior is consistent with the nonlinear Euler equations and that a similar result also holds for the nonlinear Navier–Stokes equations. Hence, these results provide an indication that nonlinear inviscid damping may still hold in Sobolev regularity in the above sense despite the Gevrey regularity instability results of Deng and Masmoudi (Long time instability of the Couette flow in low Gevrey spaces, 2018. arXiv:1803.01246).

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Notes

Acknowledgements

The author would like to thank the MPI MIS, where part of the project was written, for its hospitality.

Compliance with ethical standards

Conflict of interest

The author(s) declares that they have no competing interests.

References

  1. 1.
    Alberti, G., Crippa, G., Mazzucato, A.L.: Exponential self-similar mixing and loss of regularity for continuity equations. Comptes rendus Mathematique 352(11), 901–906 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bedrossian, J., Germain, P., Masmoudi, N.: Dynamics near the subcritical transition of the 3D Couette flow I: below threshold case (2015). arXiv preprint arXiv:1506.03720
  3. 3.
    Bedrossian, J., Germain, P., Masmoudi, N.: Dynamics near the subcritical transition of the 3D Couette flow II: above threshold case (2015). arXiv preprint arXiv:1506.03720
  4. 4.
    Bedrossian, J., Masmoudi, N.: Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations (2013). arXiv preprint arXiv:1306.5028
  5. 5.
    Bedrossian, J., Masmoudi, N.: Asymptotic stability for the Couette flow in the 2D Euler equations. Appl. Math. Res. eXpress 2014(1), 157–175 (2014)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bedrossian, J., Masmoudi, N., Mouhot, C.: Landau damping: paraproducts and Gevrey regularity. Ann. PDE 2(1), 4 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bedrossian, J., Masmoudi, N., Vicol, V.: Enhanced dissipation and inviscid damping in the inviscid limit of the Navier–Stokes equations near the two dimensional Couette flow. Arch. Ration. Mech. Anal. 219(3), 1087–1159 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bedrossian, J., Vicol, V., Wang, F.: The Sobolev stability threshold for 2D shear flows near Couette. J. Nonlinear Sci. 28, 1–25 (2016) MathSciNetzbMATHGoogle Scholar
  10. 10.
    Crippa, G., Lucà, R., Schulze, C.: Polynomial mixing under a certain stationary Euler flow (2017). arXiv preprint arXiv:1707.09909
  11. 11.
    Crippa, G., Schulze, C.: Cellular mixing with bounded palenstrophy. Math. Models Methods Appl. Sci. 27(12), 2297–2320 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Coti Zelati, M., Zillinger, C.: On degenerate circular and shear flows: the point vortex and power law circular flows (2018). arXiv preprint arXiv:1801.07371
  13. 13.
    Coti Zelati, M., Delgadino, M.G., Elgindi, T.M.: On the relation between enhanced dissipation time-scales and mixing rates (2018). arXiv preprint arXiv:1806.03258
  14. 14.
    Deng, Y., Masmoudi, N.: Long time instability of the Couette flow in low Gevrey spaces (2018). arXiv preprint arXiv:1803.01246
  15. 15.
    Ionescu, A., Jia, H.: Inviscid damping near shear flows in a channel (2018). arXiv:1808.04026
  16. 16.
    Landau, L.D.: On the vibration of the electronic plasma. J. Phys. USSR 12, 10 (1946)Google Scholar
  17. 17.
    Lin, Z., Zeng, C.: Inviscid dynamical structures near Couette Flow. Arch. Ration. Mech. Anal. 200(3), 1075–1097 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mouhot, C., Villani, C.: Landau damping. J. Math. Phys. 51(1), 015204 (2010)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Mouhot, C., Villani, C.: Landau damping. Notes de cours, CEMRACS (2010). https://cedricvillani.org/sites/dev/files/old_images/2012/08/B13.Landau.pdf
  20. 20.
    Mouhot, C., Villani, C.: On Landau damping. Acta Math. 207(1), 29–201 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Orr, W.M.F.: The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. In: Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, pp. 69–138. JSTOR (1907)Google Scholar
  22. 22.
    Rayleigh, L.: On the stability, or instability, of certain fluid motions. Proc. Lond. Math. Soc. 1(1), 57 (1879)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Stepin, S.A.: Nonself-adjoint Friedrichs model in hydrodynamic stability. Funct. Anal. Appl. 29(2), 91–101 (1995)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Wei, D., Zhang, Z.: Transition threshold for the 3D Couette flow in Sobolev space (2018). arXiv preprint arXiv:1803.01359
  25. 25.
    Wei, D., Zhang, Z., Zhao, W.: Linear inviscid damping for a class of monotone shear flow in Sobolev spaces (2015). arXiv preprint arXiv:1509.08228
  26. 26.
    Wei, D., Zhang, Z., Zhao, W.: Linear inviscid damping and vorticity depletion for shear flows (2017). arXiv preprint arXiv:1704.00428
  27. 27.
    Wei, D., Zhang, Z., Zhao, W.: Linear inviscid damping for a class of monotone shear flow in Sobolev spaces. Commun. Pure Appl. Math. 71(4), 617–687 (2018)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Zillinger, C.: Linear inviscid damping for monotone shear flows in a finite periodic channel, boundary effects, blow-up and critical Sobolev regularity. Arch. Ration. Mech. Anal. 221, 1–61 (2016)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Zillinger, C.: Linear inviscid damping for monotone shear flows. Trans. Am. Math. Soc. 369(12), 8799–8855 (2017)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Zillinger, Christian: On circular flows: linear stability and damping. J. Differ. Equ. 263, 7856–7899 (2017)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Zillinger, C.: On geometric and analytic mixing scales: comparability and convergence rates for transport problems (2018). arXiv preprint arXiv:1804.11299

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUS

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