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

Abstract

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.

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

References

  1. 1.

    V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer, New York, 1998.

    Google Scholar 

  2. 2.

    J. Baggett, T. Driscoll and L. Trefethen, A mostly linear model of transition of turbulence, Phys. Fluids, 7 (1995), 833–838.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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.

    Google Scholar 

  5. 5.

    N. Balmforth and P. Morrison, Normal modes and continuous spectra, Ann. N.Y. Acad. Sci., 773 (1995), 80–94.

    Article  Google 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).

  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.

    MathSciNet  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  MATH  Google 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.

    Article  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    J. Boyd, The continuous spectrum of linear Couette flow with the beta effect, J. Atmos. Sci., 40 (1983), 2304–2308.

    Article  Google 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).

  19. 19.

    E. Caglioti and C. Maffei, Time asymptotics for solutions of Vlasov-Poisson equation in a circle, J. Stat. Phys., 92 (1998).

  20. 20.

    R. Camassa and C. Viotti, Transient dynamics by continuous-spectrum perturbations in stratified shear flows, J. Fluid Mech., 717 (2013).

  21. 21.

    K. M. Case, Plasma oscillations, Ann. Phys., 7 (1959), 349–364.

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    K. M. Case, Stability of inviscid plane Couette flow, Phys. Fluids, 3 (1960), 143–148.

    MathSciNet  Article  Google 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.

    Article  Google 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.

    MathSciNet  Article  MATH  Google 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).

    MathSciNet  Article  MATH  Google 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.

    Article  MATH  Google Scholar 

  28. 28.

    P. Degond, Spectral theory of the linearized Vlasov-Poisson equation, Trans. Am. Math. Soc., 294 (1986), 435–453.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

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

    Google Scholar 

  31. 31.

    T. Ellingsen and E. Palm, Stability of linear flow, Phys. Fluids, 18 (1975), 487.

    Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    D. Gérard-Varet and N. Masmoudi, Well-posedness for the Prandtl system without analyticity or monotonicity, preprint (2013).

  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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Google Scholar 

  38. 38.

    A. Gilbert, Spiral structures and spectra in two-dimensional turbulence, J. Fluid Mech., 193 (1988), 475–497.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    R. Glassey and J. Schaeffer, On time decay rates in landau damping, Commun. Partial Differ. Equ., 20 (1995), 647–676.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  44. 44.

    Y. Guo and G. Rein, Isotropic steady states in galactic dynamics, Commun. Math. Phys., 219 (2001), 607–629.

    MathSciNet  Article  MATH  Google Scholar 

  45. 45.

    G. Hagstrom and P. Morrison, Caldeira-Leggett model, Landau damping and the Vlasov-Poisson system, Physica D, 240 (2011), 1652–1660.

    MathSciNet  Article  MATH  Google 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.

  47. 47.

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

    Article  Google Scholar 

  48. 48.

    R. Kraichnan, Inertial ranges in two-dimensional turbulence, Phys. Fluids, 10 (1967).

  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.

    MathSciNet  Article  MATH  Google Scholar 

  50. 50.

    S. B. Kuksin and A. Shirikyan, Mathematics of Two-Dimensional Turbulence, Cambridge University Press, Cambridge, 2012.

    Google Scholar 

  51. 51.

    L. Landau, On the vibration of the electronic plasma, J. Phys. USSR, 10 (1946).

  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.

    MathSciNet  Article  MATH  Google Scholar 

  54. 54.

    D. Levermore and M. Oliver, Analyticity of solutions for a generalized Euler equation, J. Differ. Equ., 133 (1997), 321–339.

    MathSciNet  Article  MATH  Google Scholar 

  55. 55.

    Y. Li and Z. Lin, A resolution of the Sommerfeld paradox, SIAM J. Math. Anal., 43 (2011), 1923–1954.

    MathSciNet  Article  MATH  Google Scholar 

  56. 56.

    C. C. Lin, The Theory of Hydrodynamic Stability, Cambridge University Press, Cambridge, 1955.

    Google Scholar 

  57. 57.

    Z. Lin, Nonlinear instability of ideal plane flows, Int. Math. Res. Not., 41 (2004), 2147–2178.

    Article  Google Scholar 

  58. 58.

    Z. Lin and C. Zeng, Inviscid dynamic structures near Couette flow, Arch. Ration. Mech. Anal., 200 (2011), 1075–1097.

    MathSciNet  Article  MATH  Google Scholar 

  59. 59.

    Z. Lin and C. Zeng, Small BGK waves and nonlinear Landau damping, Commun. Math. Phys., 306 (2011), 291–331.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  61. 61.

    R. Lindzen, Instability of plane parallel shear flow (toward a mechanistic picture of how it works), PAGEOPH, 126 (1988).

  62. 62.

    A. Lundbladh and A. V. Johansson, Direct simulation of turbulent spots in plane Couette flow, J. Fluid Mech., 229 (1991), 499–516.

    Article  MATH  Google 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.

    Google Scholar 

  64. 64.

    J. Malmberg and C. Wharton, Collisionless damping of electrostatic plasma waves, Phys. Rev. Lett., 13 (1964), 184–186.

    Article  Google Scholar 

  65. 65.

    J. Malmberg, C. Wharton, C. Gould and T. O’Neil, Plasma wave echo, Phys. Rev. Lett., 20 (1968), 95–97.

    Article  Google 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.

    Article  MATH  Google Scholar 

  67. 67.

    N. Masmoudi and K. Nakanishi, Energy convergence for singular limits of Zakharov type systems, Invent. Math., 172 (2008), 535–583.

    MathSciNet  Article  MATH  Google Scholar 

  68. 68.

    P. J. Morrison, Hamiltonian description of the ideal fluid, Rev. Mod. Phys., 70 (1998), 467–521.

    Article  MATH  Google 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.

    Article  MATH  Google Scholar 

  70. 70.

    C. Mouhot and C. Villani, On Landau damping, Acta Math., 207 (2011), 29–201.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  72. 72.

    L. Nirenberg, An abstract form of the nonlinear Cauchy-Kowalewski theorem, J. Differ. Geom., 6 (1972), 561–576.

    MathSciNet  MATH  Google Scholar 

  73. 73.

    T. Nishida, A note on a theorem of Nirenberg, J. Differ. Geom., 12 (1977), 629–633.

    MathSciNet  MATH  Google 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.

    Article  MATH  Google Scholar 

  76. 76.

    L. Rayleigh, On the stability, or instability, of certain fluid motions, Proc. Lond. Math. Soc., S1-11 (1880), 57.

    MathSciNet  Article  Google 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.

    MathSciNet  Article  MATH  Google 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.

    Article  Google Scholar 

  79. 79.

    D. Ryutov, Landau damping: half a century with the great discovery, Plasma Phys. Control. Fusion, 41 (1999), A1.

    Article  Google 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).

  81. 81.

    P. J. Schmid and D. S. Henningson, Stability and Transition in Shear Flows, Applied Mathematical Sciences., vol. 142, Springer, New York, 2001.

    Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  83. 83.

    A. Shnirelman, On the long time behavior of fluid flows, preprint (2012).

  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.

    MathSciNet  Article  MATH  Google Scholar 

  85. 85.

    J. Tataronis and W. Grossmann, Decay of MHD waves by phase mixing, Z. Phys., 261 (1973), 203–216.

    Article  Google Scholar 

  86. 86.

    N. Tillmark and P. Alfredsson, Experiments on transition in plane Couette flow, J. Fluid Mech., 235 (1992), 89–102.

    Article  Google Scholar 

  87. 87.

    L. Trefethen, A. Trefethen, S. Reddy and T. Driscoll, Hydrodynamic stability without eigenvalues, Science, 261 (1993), 578–584.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  90. 90.

    N. van Kampen, On the theory of stationary waves in plasmas, Physica, 21 (1955), 949–963.

    MathSciNet  Article  Google Scholar 

  91. 91.

    J. Vanneste, Nonlinear dynamics of anisotropic disturbances in plane Couette flow, SIAM J. Appl. Math., 62 (2002), 924–944 (electronic).

    MathSciNet  Article  MATH  Google 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.

    Article  Google 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).

  95. 95.

    J. Yu, C. Driscoll and T. O‘Neil, Phase mixing and echoes in a pure electron plasma, Phys. Plasmas, 12 (2005), 055701.

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Nader Masmoudi.

Additional information

J. Bedrossian was partially supported by NSF Postdoctoral Fellowship in Mathematical Sciences, DMS-1103765.

N. Masmoudi was partially supported by NSF grant DMS-1211806.

The current email and address are jacob@cscamm.umd.edu, Department of Mathematics and the Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD, USA.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bedrossian, J., Masmoudi, N. Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations. Publ.math.IHES 122, 195–300 (2015). https://doi.org/10.1007/s10240-015-0070-4

Download citation

Keywords

  • Asymptotic Stability
  • Planar Shear
  • Vlasov Equation
  • Gevrey Class
  • Background Shear