Publications mathématiques de l'IHÉS

, Volume 122, Issue 1, pp 195–300 | Cite as

Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations

  • Jacob Bedrossian
  • Nader MasmoudiEmail author


We prove asymptotic stability of shear flows close to the planar Couette flow in the 2D inviscid Euler equations on T×R. That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L 2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically driven to small scales by a linear evolution and weakly converges as t→±∞. The strong convergence of the velocity field is sometimes referred to as inviscid damping, due to the relationship with Landau damping in the Vlasov equations. This convergence was formally derived at the linear level by Kelvin in 1887 and it occurs at an algebraic rate first computed by Orr in 1907; our work appears to be the first rigorous confirmation of this behavior on the nonlinear level.


Asymptotic Stability Planar Shear Vlasov Equation Gevrey Class Background Shear 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer, New York, 1998. zbMATHGoogle Scholar
  2. 2.
    J. Baggett, T. Driscoll and L. Trefethen, A mostly linear model of transition of turbulence, Phys. Fluids, 7 (1995), 833–838. MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    H. Bahouri and J.-Y. Chemin, Équations de transport relatives á des champs de vecteurs non-lipschitziens et mécanique des fluides, Arch. Ration. Mech. Anal., 127 (1994), 159–181. MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 343, Springer, Heidelberg, 2011. zbMATHGoogle Scholar
  5. 5.
    N. Balmforth and P. Morrison, Normal modes and continuous spectra, Ann. N.Y. Acad. Sci., 773 (1995), 80–94. CrossRefGoogle Scholar
  6. 6.
    N. Balmforth and P. Morrison, Singular eigenfunctions for shearing fluids I, Institute for Fusion Studies Report, University of Texas-Austin, 692 (1995), 1–80. Google Scholar
  7. 7.
    N. Balmforth, P. Morrison and J.-L. Thiffeault, Pattern formation in Hamiltonian systems with continuous spectra; a normal-form single-wave model, preprint (2013). Google Scholar
  8. 8.
    N. J. Balmforth and P. J. Morrison, Hamiltonian description of shear flow, in Large-Scale Atmosphere-Ocean Dynamics, vol. II, pp. 117–142, Cambridge Univ. Press, Cambridge, 2002. Google Scholar
  9. 9.
    C. Bardos and S. Benachour, Domaine d’analycité des solutions de l’équation d’Euler dans un ouvert de R n, Ann. Sc. Norm. Super. Pisa, Cl. Sci., 4 (1977), 647–687. MathSciNetzbMATHGoogle Scholar
  10. 10.
    C. Bardos, Y. Guo and W. Strauss, Stable and unstable ideal plane flows, Chin. Ann. Math., Ser. B, 23 (2002), 149–164. Dedicated to the memory of Jacques-Louis Lions. MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    A. Bassom and A. Gilbert, The spiral wind-up of vorticity in an inviscid planar vortex, J. Fluid Mech., 371 (1998), 109–140. MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    J. Bedrossian, N. Masmoudi and C. Mouhot, Landau damping: paraproducts and Gevrey regularity, arXiv:1311.2870, 2013.
  13. 13.
    J. Bedrossian, N. Masmoudi and V. Vicol, Enhanced dissipation and inviscid damping in the inviscid limit of the Navier-Stokes equations near the 2D Couette flow, arXiv:1408.4754, 2014.
  14. 14.
    J. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non lináires, Ann. Sci. Éc. Norm. Super., 14 (1981), 209–246. MathSciNetzbMATHGoogle Scholar
  15. 15.
    S. Bottin, O. Dauchot, F. Daviaud and P. Manneville, Experimental evidence of streamwise vortices as finite amplitude solutions in transitional plane Couette flow, Phys. Fluids, 10 (1998), 2597. CrossRefGoogle Scholar
  16. 16.
    F. Bouchet and H. Morita, Large time behavior and asymptotic stability of the 2D Euler and linearized Euler equations, Physica D, 239 (2010), 948–966. MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    J. Boyd, The continuous spectrum of linear Couette flow with the beta effect, J. Atmos. Sci., 40 (1983), 2304–2308. CrossRefGoogle Scholar
  18. 18.
    R. Briggs, J. Daugherty and R. Levy, Role of Landau damping in crossed-field electron beams and inviscid shear flow, Phys. Fluids, 13 (1970). Google Scholar
  19. 19.
    E. Caglioti and C. Maffei, Time asymptotics for solutions of Vlasov-Poisson equation in a circle, J. Stat. Phys., 92 (1998). Google Scholar
  20. 20.
    R. Camassa and C. Viotti, Transient dynamics by continuous-spectrum perturbations in stratified shear flows, J. Fluid Mech., 717 (2013). Google Scholar
  21. 21.
    K. M. Case, Plasma oscillations, Ann. Phys., 7 (1959), 349–364. MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    K. M. Case, Stability of inviscid plane Couette flow, Phys. Fluids, 3 (1960), 143–148. MathSciNetCrossRefGoogle Scholar
  23. 23.
    A. Cerfon, J. Freidberg, F. Parra and T. Antaya, Analytic fluid theory of beam spiraling in high-intensity cyclotrons, Phys. Rev. ST Accel. Beams, 16 (2013), 024202. CrossRefGoogle Scholar
  24. 24.
    J.-Y. Chemin, Le système de Navier-Stokes incompressible soixante dix ans après Jean Leray, in Actes des Journées Mathématiques à la Mémoire de Jean Leray, Sémin. Congr, vol. 9, pp. 99–123, Soc. Math. France, Paris, 2004. Google Scholar
  25. 25.
    J.-Y. Chemin, I. Gallagher and M. Paicu, Global regularity for some classes of large solutions to the Navier-Stokes equations, Ann. Math. (2), 173 (2011), 983–1012. MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    J.-Y. Chemin and N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., 33 (2001), 84–112 (electronic). MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    P. Constantin, A. Kiselev, L. Ryzhik and A. Zlatoš, Diffusion and mixing in fluid flow, Ann. Math. (2), 168 (2008), 643–674. CrossRefzbMATHGoogle Scholar
  28. 28.
    P. Degond, Spectral theory of the linearized Vlasov-Poisson equation, Trans. Am. Math. Soc., 294 (1986), 435–453. MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    S. A. Denisov, Infinite superlinear growth of the gradient for the two-dimensional Euler equation, Discrete Contin. Dyn. Syst., 23 (2009), 755–764. MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    P. G. Drazin and W. H. Reid, Hydrodynamic Stability, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, Cambridge, 1981. zbMATHGoogle Scholar
  31. 31.
    T. Ellingsen and E. Palm, Stability of linear flow, Phys. Fluids, 18 (1975), 487. CrossRefzbMATHGoogle Scholar
  32. 32.
    A. B. Ferrari and E. S. Titi, Gevrey regularity for nonlinear analytic parabolic equations, Commun. Partial Differ. Equ., 23 (1998), 1–16. MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    C. Foias and R. Temam, Gevrey class regularity for solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359–369. MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    S. Friedlander, W. Strauss and M. Vishik, Nonlinear instability in an ideal fluid, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 14 (1997), 187–209. MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    D. Gérard-Varet and N. Masmoudi, Well-posedness for the Prandtl system without analyticity or monotonicity, preprint (2013). Google Scholar
  36. 36.
    P. Germain, N. Masmoudi and J. Shatah, Global solutions for the gravity water waves equation in dimension 3, Ann. Math. (2), 175 (2012), 691–754. MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    M. Gevrey, Sur la nature analytique des solutions des équations aux dérivées partielles. Premier mémoire, Ann. Sci. Éc. Norm. Super., 3 (1918), 129–190. MathSciNetGoogle Scholar
  38. 38.
    A. Gilbert, Spiral structures and spectra in two-dimensional turbulence, J. Fluid Mech., 193 (1988), 475–497. MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations. I, Rev. Math. Phys., 12 (2000), 361–429. MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    R. Glassey and J. Schaeffer, Time decay for solutions to the linearized Vlasov equation, Transp. Theory Stat. Phys., 23 (1994), 411–453. MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    R. Glassey and J. Schaeffer, On time decay rates in landau damping, Commun. Partial Differ. Equ., 20 (1995), 647–676. MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    N. Glatt-Holtz, V. Sverak and V. Vicol, On inviscid limits for the stochastic Navier-Stokes equations and related models, arXiv:1302.0542, 2013.
  43. 43.
    E. Grenier, On the nonlinear instability of Euler and Prandtl equations, Commun. Pure Appl. Math., 53 (2000), 1067–1091. MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Y. Guo and G. Rein, Isotropic steady states in galactic dynamics, Commun. Math. Phys., 219 (2001), 607–629. MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    G. Hagstrom and P. Morrison, Caldeira-Leggett model, Landau damping and the Vlasov-Poisson system, Physica D, 240 (2011), 1652–1660. MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    H. Hwang and J. Velaźquez, On the existence of exponentially decreasing solutions of the nonlinear Landau damping problem, Indiana Univ. Math. J., (2009), 2623–2660. Google Scholar
  47. 47.
    L. Kelvin, Stability of fluid motion-rectilinear motion of viscous fluid between two parallel plates, Philos. Mag., 24 (1887), 188. CrossRefGoogle Scholar
  48. 48.
    R. Kraichnan, Inertial ranges in two-dimensional turbulence, Phys. Fluids, 10 (1967). Google Scholar
  49. 49.
    I. Kukavica and V. Vicol, On the radius of analyticity of solutions to the three-dimensional Euler equations, Proc. Am. Math. Soc., 137 (2009), 669–677. MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    S. B. Kuksin and A. Shirikyan, Mathematics of Two-Dimensional Turbulence, Cambridge University Press, Cambridge, 2012. CrossRefzbMATHGoogle Scholar
  51. 51.
    L. Landau, On the vibration of the electronic plasma, J. Phys. USSR, 10 (1946). Google Scholar
  52. 52.
    P. D. Lax and R. S. Phillips, Scattering Theory, vol. 26, Academic Press, San Diego, 1990. Google Scholar
  53. 53.
    M. Lemou, F. Méhats and P. Raphaël, Orbital stability of spherical galactic models, Invent. Math., 187 (2012), 145–194. MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    D. Levermore and M. Oliver, Analyticity of solutions for a generalized Euler equation, J. Differ. Equ., 133 (1997), 321–339. MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Y. Li and Z. Lin, A resolution of the Sommerfeld paradox, SIAM J. Math. Anal., 43 (2011), 1923–1954. MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    C. C. Lin, The Theory of Hydrodynamic Stability, Cambridge University Press, Cambridge, 1955. zbMATHGoogle Scholar
  57. 57.
    Z. Lin, Nonlinear instability of ideal plane flows, Int. Math. Res. Not., 41 (2004), 2147–2178. CrossRefGoogle Scholar
  58. 58.
    Z. Lin and C. Zeng, Inviscid dynamic structures near Couette flow, Arch. Ration. Mech. Anal., 200 (2011), 1075–1097. MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Z. Lin and C. Zeng, Small BGK waves and nonlinear Landau damping, Commun. Math. Phys., 306 (2011), 291–331. MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    H. Lindblad and I. Rodnianski, Global existence for the Einstein vacuum equations in wave coordinates, Commun. Math. Phys., 256 (2005), 43–110. MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    R. Lindzen, Instability of plane parallel shear flow (toward a mechanistic picture of how it works), PAGEOPH, 126 (1988). Google Scholar
  62. 62.
    A. Lundbladh and A. V. Johansson, Direct simulation of turbulent spots in plane Couette flow, J. Fluid Mech., 229 (1991), 499–516. CrossRefzbMATHGoogle Scholar
  63. 63.
    A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, vol. 9, New York University Courant Institute of Mathematical Sciences, New York, 2003. zbMATHGoogle Scholar
  64. 64.
    J. Malmberg and C. Wharton, Collisionless damping of electrostatic plasma waves, Phys. Rev. Lett., 13 (1964), 184–186. CrossRefGoogle Scholar
  65. 65.
    J. Malmberg, C. Wharton, C. Gould and T. O’Neil, Plasma wave echo, Phys. Rev. Lett., 20 (1968), 95–97. CrossRefGoogle Scholar
  66. 66.
    P. S. Marcus and W. H. Press, On Green’s functions for small disturbances of plane Couette flow, J. Fluid Mech., 79 (1977), 525–534. CrossRefzbMATHGoogle Scholar
  67. 67.
    N. Masmoudi and K. Nakanishi, Energy convergence for singular limits of Zakharov type systems, Invent. Math., 172 (2008), 535–583. MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    P. J. Morrison, Hamiltonian description of the ideal fluid, Rev. Mod. Phys., 70 (1998), 467–521. CrossRefzbMATHGoogle Scholar
  69. 69.
    P. J. Morrison, Hamiltonian description of Vlasov dynamics: action-angle variables for the continuous spectrum, Transp. Theory Stat. Phys., 29 (2000), 397–414. CrossRefzbMATHGoogle Scholar
  70. 70.
    C. Mouhot and C. Villani, On Landau damping, Acta Math., 207 (2011), 29–201. MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    K. Nakanishi, Modified wave operators for the Hartree equation with data, image and convergence in the same space, Commun. Pure Appl. Anal., 1 (2002), 237–252. MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    L. Nirenberg, An abstract form of the nonlinear Cauchy-Kowalewski theorem, J. Differ. Geom., 6 (1972), 561–576. MathSciNetzbMATHGoogle Scholar
  73. 73.
    T. Nishida, A note on a theorem of Nirenberg, J. Differ. Geom., 12 (1977), 629–633. MathSciNetzbMATHGoogle Scholar
  74. 74.
    W. Orr, 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., A Math. Phys. Sci., 27 (1907), 9–68. Google Scholar
  75. 75.
    S. A. Orszag and L. C. Kells, Transition to turbulence in plane Poiseuille and plane Couette flow, J. Fluid Mech., 96 (1980), 159–205. CrossRefzbMATHGoogle Scholar
  76. 76.
    L. Rayleigh, On the stability, or instability, of certain fluid motions, Proc. Lond. Math. Soc., S1-11 (1880), 57. MathSciNetCrossRefGoogle Scholar
  77. 77.
    S. C. Reddy, P. J. Schmid and D. S. Henningson, Pseudospectra of the Orr-Sommerfeld operator, SIAM J. Appl. Math., 53 (1993), 15–47. MathSciNetCrossRefzbMATHGoogle Scholar
  78. 78.
    O. Reynolds, An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels, Proc. R. Soc. Lond., 35 (1883), 84. CrossRefGoogle Scholar
  79. 79.
    D. Ryutov, Landau damping: half a century with the great discovery, Plasma Phys. Control. Fusion, 41 (1999), A1. CrossRefGoogle Scholar
  80. 80.
    D. Schecter, D. Dubin, A. Cass, C. Driscoll and I.L. et al., Inviscid damping of asymmetries on a two-dimensional vortex, Phys. Fluids, 12 (2000). Google Scholar
  81. 81.
    P. J. Schmid and D. S. Henningson, Stability and Transition in Shear Flows, Applied Mathematical Sciences., vol. 142, Springer, New York, 2001. zbMATHGoogle Scholar
  82. 82.
    J. Shatah and C. Zeng, Geometry and a priori estimates for free boundary problems of the Euler equation, Commun. Pure Appl. Math., 61 (2008), 698–744. MathSciNetCrossRefzbMATHGoogle Scholar
  83. 83.
    A. Shnirelman, On the long time behavior of fluid flows, preprint (2012). Google Scholar
  84. 84.
    S. Strogatz, R. Mirollow and P. Matthews, Coupled nonlinear oscillators below the synchronization threshold: relaxation by generalized Landau damping, Phys. Rev. Lett., 68 (1992), 2730–2733. MathSciNetCrossRefzbMATHGoogle Scholar
  85. 85.
    J. Tataronis and W. Grossmann, Decay of MHD waves by phase mixing, Z. Phys., 261 (1973), 203–216. CrossRefGoogle Scholar
  86. 86.
    N. Tillmark and P. Alfredsson, Experiments on transition in plane Couette flow, J. Fluid Mech., 235 (1992), 89–102. CrossRefGoogle Scholar
  87. 87.
    L. Trefethen, A. Trefethen, S. Reddy and T. Driscoll, Hydrodynamic stability without eigenvalues, Science, 261 (1993), 578–584. MathSciNetCrossRefzbMATHGoogle Scholar
  88. 88.
    L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press, Princeton, 2005. Google Scholar
  89. 89.
    K. Tung, Initial-value problems for Rossby waves in a shear flow with critical level, J. Fluid Mech., 133 (1983), 443–469. MathSciNetCrossRefzbMATHGoogle Scholar
  90. 90.
    N. van Kampen, On the theory of stationary waves in plasmas, Physica, 21 (1955), 949–963. MathSciNetCrossRefGoogle Scholar
  91. 91.
    J. Vanneste, Nonlinear dynamics of anisotropic disturbances in plane Couette flow, SIAM J. Appl. Math., 62 (2002), 924–944 (electronic). MathSciNetCrossRefzbMATHGoogle Scholar
  92. 92.
    J. Vanneste, P. Morrison and T. Warn, Strong echo effect and nonlinear transient growth in shear flows, Phys. Fluids, 10 (1998), 1398. CrossRefGoogle Scholar
  93. 93.
    A. Yaglom, Hydrodynamic Instability and Transition to Turbulence, vol. 100, Springer, Berlin, 2012. Google Scholar
  94. 94.
    J. Yu and C. Driscoll, Diocotron wave echoes in a pure electron plasma, IEEE Trans. Plasma Sci., 30 (2002). Google Scholar
  95. 95.
    J. Yu, C. Driscoll and T. O‘Neil, Phase mixing and echoes in a pure electron plasma, Phys. Plasmas, 12 (2005), 055701. CrossRefGoogle Scholar

Copyright information

© IHES and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

Personalised recommendations