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

Bedrossian, Jacob; Masmoudi, Nader

Publications Mathématiques de l'IHÉS, Tome 122 (2015), pp. 195-300.

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 L² 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.

Mots clés : Asymptotic Stability, Planar Shear, Vlasov Equation, Gevrey Class, Background Shear

[1.] Arnold, V. I.; Khesin, B. A. Topological Methods in Hydrodynamics (1998) | Zbl

[2.] Baggett, J.; Driscoll, T.; Trefethen, L. A mostly linear model of transition of turbulence, Phys. Fluids, Volume 7 (1995), pp. 833-838 | DOI | MR | Zbl

[3.] Bahouri, H.; Chemin, J.-Y. Équations de transport relatives á des champs de vecteurs non-lipschitziens et mécanique des fluides, Arch. Ration. Mech. Anal., Volume 127 (1994), pp. 159-181 | DOI | MR | Zbl

[4.] Bahouri, H.; Chemin, J.-Y.; Danchin, R. Fourier Analysis and Nonlinear Partial Differential Equations (2011) | Zbl

[5.] Balmforth, N.; Morrison, P. Normal modes and continuous spectra, Ann. N.Y. Acad. Sci., Volume 773 (1995), pp. 80-94 | DOI

[6.] Balmforth, N.; Morrison, P. Singular eigenfunctions for shearing fluids I, Institute for Fusion Studies Report, University of Texas-Austin, Volume 692 (1995), pp. 1-80

[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.] Balmforth, N. J.; Morrison, P. J. Hamiltonian description of shear flow, Large-Scale Atmosphere-Ocean Dynamics (2002), pp. 117-142

[9.] Bardos, C.; Benachour, S. Domaine d’analycité des solutions de l’équation d’Euler dans un ouvert de Rⁿ, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 4 (1977), pp. 647-687 | Numdam | MR | Zbl

[10.] Bardos, C.; Guo, Y.; Strauss, W. Stable and unstable ideal plane flows, Chin. Ann. Math., Ser. B, Volume 23 (2002), pp. 149-164 (Dedicated to the memory of Jacques-Louis Lions) | DOI | MR | Zbl

[11.] Bassom, A.; Gilbert, A. The spiral wind-up of vorticity in an inviscid planar vortex, J. Fluid Mech., Volume 371 (1998), pp. 109-140 | DOI | MR | Zbl

[12.] J. Bedrossian, N. Masmoudi and C. Mouhot, Landau damping: paraproducts and Gevrey regularity, | arXiv

[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

[14.] Bony, J. Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non lináires, Ann. Sci. Éc. Norm. Super., Volume 14 (1981), pp. 209-246 | Numdam | MR | Zbl

[15.] Bottin, S.; Dauchot, O.; Daviaud, F.; Manneville, P. Experimental evidence of streamwise vortices as finite amplitude solutions in transitional plane Couette flow, Phys. Fluids, Volume 10 (1998), p. 2597 | DOI

[16.] Bouchet, F.; Morita, H. Large time behavior and asymptotic stability of the 2D Euler and linearized Euler equations, Physica D, Volume 239 (2010), pp. 948-966 | DOI | MR | Zbl

[17.] Boyd, J. The continuous spectrum of linear Couette flow with the beta effect, J. Atmos. Sci., Volume 40 (1983), pp. 2304-2308 | DOI

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

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

[21.] Case, K. M. Plasma oscillations, Ann. Phys., Volume 7 (1959), pp. 349-364 | DOI | MR | Zbl

[22.] Case, K. M. Stability of inviscid plane Couette flow, Phys. Fluids, Volume 3 (1960), pp. 143-148 | DOI | MR

[23.] Cerfon, A.; Freidberg, J.; Parra, F.; Antaya, T. Analytic fluid theory of beam spiraling in high-intensity cyclotrons, Phys. Rev. ST Accel. Beams, Volume 16 (2013) | DOI

[24.] Chemin, J.-Y. Le système de Navier-Stokes incompressible soixante dix ans après Jean Leray, Actes des Journées Mathématiques à la Mémoire de Jean Leray (2004), pp. 99-123

[25.] Chemin, J.-Y.; Gallagher, I.; Paicu, M. Global regularity for some classes of large solutions to the Navier-Stokes equations, Ann. Math. (2), Volume 173 (2011), pp. 983-1012 | DOI | MR | Zbl

[26.] Chemin, J.-Y.; Masmoudi, N. About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., Volume 33 (2001), pp. 84-112 (electronic) | DOI | MR | Zbl

[27.] Constantin, P.; Kiselev, A.; Ryzhik, L.; Zlatoš, A. Diffusion and mixing in fluid flow, Ann. Math. (2), Volume 168 (2008), pp. 643-674 | DOI | Zbl

[28.] Degond, P. Spectral theory of the linearized Vlasov-Poisson equation, Trans. Am. Math. Soc., Volume 294 (1986), pp. 435-453 | DOI | MR | Zbl

[29.] Denisov, S. A. Infinite superlinear growth of the gradient for the two-dimensional Euler equation, Discrete Contin. Dyn. Syst., Volume 23 (2009), pp. 755-764 | DOI | MR | Zbl

[30.] Drazin, P. G.; Reid, W. H. Hydrodynamic Stability (1981) | Zbl

[31.] Ellingsen, T.; Palm, E. Stability of linear flow, Phys. Fluids, Volume 18 (1975), p. 487 | DOI | Zbl

[32.] Ferrari, A. B.; Titi, E. S. Gevrey regularity for nonlinear analytic parabolic equations, Commun. Partial Differ. Equ., Volume 23 (1998), pp. 1-16 | DOI | MR | Zbl

[33.] Foias, C.; Temam, R. Gevrey class regularity for solutions of the Navier-Stokes equations, J. Funct. Anal., Volume 87 (1989), pp. 359-369 | DOI | MR | Zbl

[34.] Friedlander, S.; Strauss, W.; Vishik, M. Nonlinear instability in an ideal fluid, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 14 (1997), pp. 187-209 | DOI | Numdam | MR | Zbl

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

[36.] Germain, P.; Masmoudi, N.; Shatah, J. Global solutions for the gravity water waves equation in dimension 3, Ann. Math. (2), Volume 175 (2012), pp. 691-754 | DOI | MR | Zbl

[37.] Gevrey, M. Sur la nature analytique des solutions des équations aux dérivées partielles. Premier mémoire, Ann. Sci. Éc. Norm. Super., Volume 3 (1918), pp. 129-190 | MR

[38.] Gilbert, A. Spiral structures and spectra in two-dimensional turbulence, J. Fluid Mech., Volume 193 (1988), pp. 475-497 | DOI | MR | Zbl

[39.] Ginibre, J.; Velo, G. Long range scattering and modified wave operators for some Hartree type equations. I, Rev. Math. Phys., Volume 12 (2000), pp. 361-429 | DOI | MR | Zbl

[40.] Glassey, R.; Schaeffer, J. Time decay for solutions to the linearized Vlasov equation, Transp. Theory Stat. Phys., Volume 23 (1994), pp. 411-453 | DOI | MR | Zbl

[41.] Glassey, R.; Schaeffer, J. On time decay rates in landau damping, Commun. Partial Differ. Equ., Volume 20 (1995), pp. 647-676 | DOI | MR | Zbl

[42.] N. Glatt-Holtz, V. Sverak and V. Vicol, On inviscid limits for the stochastic Navier-Stokes equations and related models, | arXiv

[43.] Grenier, E. On the nonlinear instability of Euler and Prandtl equations, Commun. Pure Appl. Math., Volume 53 (2000), pp. 1067-1091 | DOI | MR | Zbl

[44.] Guo, Y.; Rein, G. Isotropic steady states in galactic dynamics, Commun. Math. Phys., Volume 219 (2001), pp. 607-629 | DOI | MR | Zbl

[45.] Hagstrom, G.; Morrison, P. Caldeira-Leggett model, Landau damping and the Vlasov-Poisson system, Physica D, Volume 240 (2011), pp. 1652-1660 | DOI | MR | Zbl

[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.] Kelvin, L. Stability of fluid motion-rectilinear motion of viscous fluid between two parallel plates, Philos. Mag., Volume 24 (1887), p. 188 | DOI

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

[49.] Kukavica, I.; Vicol, V. On the radius of analyticity of solutions to the three-dimensional Euler equations, Proc. Am. Math. Soc., Volume 137 (2009), pp. 669-677 | DOI | MR | Zbl

[50.] Kuksin, S. B.; Shirikyan, A. Mathematics of Two-Dimensional Turbulence (2012) | DOI | Zbl

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

[52.] Lax, P. D.; Phillips, R. S. Scattering Theory (1990)

[53.] Lemou, M.; Méhats, F.; Raphaël, P. Orbital stability of spherical galactic models, Invent. Math., Volume 187 (2012), pp. 145-194 | DOI | MR | Zbl

[54.] Levermore, D.; Oliver, M. Analyticity of solutions for a generalized Euler equation, J. Differ. Equ., Volume 133 (1997), pp. 321-339 | DOI | MR | Zbl

[55.] Li, Y.; Lin, Z. A resolution of the Sommerfeld paradox, SIAM J. Math. Anal., Volume 43 (2011), pp. 1923-1954 | DOI | MR | Zbl

[56.] Lin, C. C. The Theory of Hydrodynamic Stability (1955) | Zbl

[57.] Lin, Z. Nonlinear instability of ideal plane flows, Int. Math. Res. Not., Volume 41 (2004), pp. 2147-2178 | DOI

[58.] Lin, Z.; Zeng, C. Inviscid dynamic structures near Couette flow, Arch. Ration. Mech. Anal., Volume 200 (2011), pp. 1075-1097 | DOI | MR | Zbl

[59.] Lin, Z.; Zeng, C. Small BGK waves and nonlinear Landau damping, Commun. Math. Phys., Volume 306 (2011), pp. 291-331 | DOI | MR | Zbl

[60.] Lindblad, H.; Rodnianski, I. Global existence for the Einstein vacuum equations in wave coordinates, Commun. Math. Phys., Volume 256 (2005), pp. 43-110 | DOI | MR | Zbl

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

[62.] Lundbladh, A.; Johansson, A. V. Direct simulation of turbulent spots in plane Couette flow, J. Fluid Mech., Volume 229 (1991), pp. 499-516 | DOI | Zbl

[63.] Majda, A. Introduction to PDEs and Waves for the Atmosphere and Ocean (2003) | Zbl

[64.] Malmberg, J.; Wharton, C. Collisionless damping of electrostatic plasma waves, Phys. Rev. Lett., Volume 13 (1964), pp. 184-186 | DOI

[65.] Malmberg, J.; Wharton, C.; Gould, C.; O’Neil, T. Plasma wave echo, Phys. Rev. Lett., Volume 20 (1968), pp. 95-97 | DOI

[66.] Marcus, P. S.; Press, W. H. On Green’s functions for small disturbances of plane Couette flow, J. Fluid Mech., Volume 79 (1977), pp. 525-534 | DOI | Zbl

[67.] Masmoudi, N.; Nakanishi, K. Energy convergence for singular limits of Zakharov type systems, Invent. Math., Volume 172 (2008), pp. 535-583 | DOI | MR | Zbl

[68.] Morrison, P. J. Hamiltonian description of the ideal fluid, Rev. Mod. Phys., Volume 70 (1998), pp. 467-521 | DOI | Zbl

[69.] Morrison, P. J. Hamiltonian description of Vlasov dynamics: action-angle variables for the continuous spectrum, Transp. Theory Stat. Phys., Volume 29 (2000), pp. 397-414 | DOI | Zbl

[70.] Mouhot, C.; Villani, C. On Landau damping, Acta Math., Volume 207 (2011), pp. 29-201 | DOI | MR | Zbl

[71.] Nakanishi, K. Modified wave operators for the Hartree equation with data, image and convergence in the same space, Commun. Pure Appl. Anal., Volume 1 (2002), pp. 237-252 | DOI | MR | Zbl

[72.] Nirenberg, L. An abstract form of the nonlinear Cauchy-Kowalewski theorem, J. Differ. Geom., Volume 6 (1972), pp. 561-576 | MR | Zbl

[73.] Nishida, T. A note on a theorem of Nirenberg, J. Differ. Geom., Volume 12 (1977), pp. 629-633 | MR | Zbl

[74.] 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., A Math. Phys. Sci., Volume 27 (1907), pp. 9-68

[75.] Orszag, S. A.; Kells, L. C. Transition to turbulence in plane Poiseuille and plane Couette flow, J. Fluid Mech., Volume 96 (1980), pp. 159-205 | DOI | Zbl

[76.] Rayleigh, L. On the stability, or instability, of certain fluid motions, Proc. Lond. Math. Soc., Volume S1-11 (1880), p. 57 | DOI | MR

[77.] Reddy, S. C.; Schmid, P. J.; Henningson, D. S. Pseudospectra of the Orr-Sommerfeld operator, SIAM J. Appl. Math., Volume 53 (1993), pp. 15-47 | DOI | MR | Zbl

[78.] Reynolds, O. 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., Volume 35 (1883), p. 84 | DOI

[79.] Ryutov, D. Landau damping: half a century with the great discovery, Plasma Phys. Control. Fusion, Volume 41 (1999) | DOI

[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.] Schmid, P. J.; Henningson, D. S. Stability and Transition in Shear Flows (2001) | Zbl

[82.] Shatah, J.; Zeng, C. Geometry and a priori estimates for free boundary problems of the Euler equation, Commun. Pure Appl. Math., Volume 61 (2008), pp. 698-744 | DOI | MR | Zbl

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

[84.] Strogatz, S.; Mirollow, R.; Matthews, P. Coupled nonlinear oscillators below the synchronization threshold: relaxation by generalized Landau damping, Phys. Rev. Lett., Volume 68 (1992), pp. 2730-2733 | DOI | MR | Zbl

[85.] Tataronis, J.; Grossmann, W. Decay of MHD waves by phase mixing, Z. Phys., Volume 261 (1973), pp. 203-216 | DOI

[86.] Tillmark, N.; Alfredsson, P. Experiments on transition in plane Couette flow, J. Fluid Mech., Volume 235 (1992), pp. 89-102 | DOI

[87.] Trefethen, L.; Trefethen, A.; Reddy, S.; Driscoll, T. Hydrodynamic stability without eigenvalues, Science, Volume 261 (1993), pp. 578-584 | DOI | MR | Zbl

[88.] Trefethen, L. N.; Embree, M. Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators (2005)

[89.] Tung, K. Initial-value problems for Rossby waves in a shear flow with critical level, J. Fluid Mech., Volume 133 (1983), pp. 443-469 | DOI | MR | Zbl

[90.] van Kampen, N. On the theory of stationary waves in plasmas, Physica, Volume 21 (1955), pp. 949-963 | DOI | MR

[91.] Vanneste, J. Nonlinear dynamics of anisotropic disturbances in plane Couette flow, SIAM J. Appl. Math., Volume 62 (2002), pp. 924-944 (electronic) | DOI | MR | Zbl

[92.] Vanneste, J.; Morrison, P.; Warn, T. Strong echo effect and nonlinear transient growth in shear flows, Phys. Fluids, Volume 10 (1998), p. 1398 | DOI

[93.] Yaglom, A. Hydrodynamic Instability and Transition to Turbulence (2012)

[94.] J. Yu and C. Driscoll, Diocotron wave echoes in a pure electron plasma, IEEE Trans. Plasma Sci., 30 (2002).

[95.] Yu, J.; Driscoll, C.; O‘Neil, T. Phase mixing and echoes in a pure electron plasma, Phys. Plasmas, Volume 12 (2005) | DOI