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Stability of the Couette flow at high Reynolds numbers in two dimensions and three dimensions
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by Jacob Bedrossian, Pierre Germain and Nader Masmoudi PDF
Bull. Amer. Math. Soc. 56 (2019), 373-414 Request permission

Abstract:

We review works on the asymptotic stability of the Couette flow. The majority of this paper is aimed toward a wide range of applied mathematicians, and there is an additional section aimed toward experts in the mathematical analysis of PDEs.
References
  • F. Alavyoon, D. S. Henningson, and P. H. Alfredsson, Turbulent spots in plane Poiseuille flow–flow visualization, The Physics of Fluids, 29 (1986), no. 4, 1328–1331.
  • S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math. 145 (2001), no. 3, 597–618. MR 1856402, DOI 10.1007/s002220100165
  • V. I. Arnol′d, On conditions for non-linear stability of plane stationary curvilinear flows of an ideal fluid, Dokl. Akad. Nauk SSSR 162 (1965), 975–978 (Russian). MR 0180051
  • Jeffrey S. Baggett, Tobin A. Driscoll, and Lloyd N. Trefethen, A mostly linear model of transition to turbulence, Phys. Fluids 7 (1995), no. 4, 833–838. MR 1324952, DOI 10.1063/1.868606
  • Jeffrey S. Baggett and Lloyd N. Trefethen, Low-dimensional models of subcritical transition to turbulence, Phys. Fluids 9 (1997), no. 4, 1043–1053. MR 1437563, DOI 10.1063/1.869199
  • Andrew P. Bassom and Andrew D. Gilbert, The spiral wind-up of vorticity in an inviscid planar vortex, J. Fluid Mech. 371 (1998), 109–140. MR 1650153, DOI 10.1017/S0022112098001955
  • K. Beauchard and E. Zuazua, Some controllability results for the 2D Kolmogorov equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 5, 1793–1815. MR 2566710, DOI 10.1016/j.anihpc.2008.12.005
  • Margaret Beck and C. Eugene Wayne, Metastability and rapid convergence to quasi-stationary bar states for the two-dimensional Navier-Stokes equations, Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), no. 5, 905–927. MR 3109765, DOI 10.1017/S0308210511001478
  • J. Bedrossian, Nonlinear echoes and Landau damping with insufficient regularity, arXiv:1605.06841 (2016).
  • Jacob Bedrossian, Suppression of plasma echoes and Landau damping in Sobolev spaces by weak collisions in a Vlasov-Fokker-Planck equation, Ann. PDE 3 (2017), no. 2, Paper No. 19, 66. MR 3719106, DOI 10.1007/s40818-017-0036-6
  • J. Bedrossian, P. Germain, and N. Masmoudi, Dynamics near the subcritical transition of the 3D Couette flow I: Below threshold, arXiv:1506.03720 (2015); Mem. Amer. Math. Soc. (to appear).
  • J. Bedrossian, P. Germain, and N. Masmoudi, Dynamics near the subcritical transition of the 3D Couette flow II: Above threshold, arXiv:1506.03721 (2015).
  • Jacob Bedrossian, Pierre Germain, and Nader Masmoudi, On the stability threshold for the 3D Couette flow in Sobolev regularity, Ann. of Math. (2) 185 (2017), no. 2, 541–608. MR 3612004, DOI 10.4007/annals.2017.185.2.4
  • J. Bedrossian and N. Masmoudi, Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations, Publ. Math. de l’IHÉS 122 (2015), no. 1, 195–300.
  • Jacob Bedrossian, Nader Masmoudi, and Clément Mouhot, Landau damping in finite regularity for unconfined systems with screened interactions, Comm. Pure Appl. Math. 71 (2018), no. 3, 537–576. MR 3762277, DOI 10.1002/cpa.21730
  • Jacob Bedrossian, Nader Masmoudi, and Clément Mouhot, Landau damping: paraproducts and Gevrey regularity, Ann. PDE 2 (2016), no. 1, Art. 4, 71. MR 3489904, DOI 10.1007/s40818-016-0008-2
  • Jacob Bedrossian, Nader Masmoudi, and Vlad Vicol, 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 (2016), no. 3, 1087–1159. MR 3448924, DOI 10.1007/s00205-015-0917-3
  • J. Bedrossian, V. Vicol, and F. Wang, The Sobolev stability threshold for 2D shear flows near Couette, arXiv:1604.01831 (2016); J. Nonlin. Sci. (to appear).
  • Jacob Bedrossian and Michele Coti Zelati, Enhanced dissipation, hypoellipticity, and anomalous small noise inviscid limits in shear flows, Arch. Ration. Mech. Anal. 224 (2017), no. 3, 1161–1204. MR 3621820, DOI 10.1007/s00205-017-1099-y
  • Jacob Bedrossian, Michele Coti Zelati, and Nathan Glatt-Holtz, Invariant measures for passive scalars in the small noise inviscid limit, Comm. Math. Phys. 348 (2016), no. 1, 101–127. MR 3551262, DOI 10.1007/s00220-016-2758-9
  • J. Bedrossian, M. C. Zelati, and V. Vicol, Vortex axisymmetrization, inviscid damping, and vorticity depletion in the linearized 2d Euler equations, arXiv:1711.03668 (2017).
  • A. Bernoff and J. Lingevitch, Rapid relaxation of an axisymmetric vortex, Phys. Fluids, 6 (1994), 3717.
  • Jean-Michel Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 2, 209–246 (French). MR 631751
  • S. Bottin, O. Dauchot, F. Daviaud, and P. Manneville, Experimental evidence of streamwise vortices as finite amplitude solutions in transitional plane Couette flow, Physics of Fluids, 10 (1998), 2597.
  • J. P. Boyd, The continuous spectrum of linear Couette flow with the beta effect, J. Atmospheric Sciences, 40 (1983), no. 9, 2304–2308.
  • T. J. M. Boyd and J. J. Sanderson, The physics of plasmas, Cambridge University Press, Cambridge, 2003. MR 1960956, DOI 10.1017/CBO9780511755750
  • E. Caglioti and C. Maffei, Time asymptotics for solutions of Vlasov-Poisson equation in a circle, J. Statist. Phys. 92 (1998), no. 1-2, 301–323. MR 1645659, DOI 10.1023/A:1023055905124
  • D. R. Carlson, S. E. Widnall, and M. F. Peeters, A flow-visualization study of transition in plane Poiseuille flow, Journal of Fluid Mechanics, 121 (1982), 487–505.
  • K. M. Case, Stability of inviscid plane Couette flow, Phys. Fluids 3 (1960), 143–148. MR 128230, DOI 10.1063/1.1706010
  • A. Castro, D. Córdoba, and J. Gómez-Serrano, Uniformly rotating smooth solutions for the incompressible 2d Euler equations, arXiv:1612.08964 (2016).
  • 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.
  • S. Jonathan Chapman, Subcritical transition in channel flows, J. Fluid Mech. 451 (2002), 35–97. MR 1886008, DOI 10.1017/S0022112001006255
  • P. Constantin, A. Kiselev, L. Ryzhik, and A. Zlatoš, Diffusion and mixing in fluid flow, Ann. of Math. (2) 168 (2008), no. 2, 643–674. MR 2434887, DOI 10.4007/annals.2008.168.643
  • A. D. Craik, Nonlinear resonant instability in boundary layers, Journal of Fluid Mechanics, 50 (1971), no. 2, 393–413.
  • F. Daviaud, J. Hegseth, and P. Bergé, Subcritical transition to turbulence in plane Couette flow, Phys. Rev. Lett. 69 (1992), no. 17, 2511.
  • Pierre Degond, Spectral theory of the linearized Vlasov-Poisson equation, Trans. Amer. Math. Soc. 294 (1986), no. 2, 435–453. MR 825714, DOI 10.1090/S0002-9947-1986-0825714-8
  • Y. Deng and N. Masmoudi, Long time instability of the Couette flow in low Gevrey spaces, arXiv:1803.01246 (2018).
  • Helge Dietert, Stability and bifurcation for the Kuramoto model, J. Math. Pures Appl. (9) 105 (2016), no. 4, 451–489 (English, with English and French summaries). MR 3471147, DOI 10.1016/j.matpur.2015.11.001
  • L. Dikiĭ, The stability of plane-parallel flows of an ideal fluid, Soviet Physics Doklady 5, (1961), 1179.
  • P. Drazin and L. Howard, Hydrodynamic stability of parallel flow of inviscid fluid, Advances in Applied Mechanics 9 (1966), 1–89.
  • P. G. Drazin and W. H. Reid, Hydrodynamic stability, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, Cambridge-New York, 1982. MR 684214
  • D. G. Dritschel, Nonlinear stability bounds for inviscid, two-dimensional, parallel or circular flows with monotonic vorticity, and the analogous three-dimensional quasi-geostrophic flows, Journal of Fluid Mechanics 191 (1988), 575–581.
  • B. Dubrulle and S. Nazarenko, On scaling laws for the transition to turbulence in uniform-shear flows, Euro. Phys. Lett. 27 (1994), no. 2, 129.
  • Yohann Duguet, Luca Brandt, and B. Robin J. Larsson, Towards minimal perturbations in transitional plane Couette flow, Phys. Rev. E (3) 82 (2010), no. 2, 026316, 13. MR 2736445, DOI 10.1103/PhysRevE.82.026316
  • Y. Duguet, P. Schlatter, and D. S. Henningson, Formation of turbulent patterns near the onset of transition in plane Couette flow, J. of Fluid Mech. 650 (2010), 119–129.
  • T. Ellingsen and E. Palm, Stability of linear flow, Phys. of Fluids 18 (1975), 487.
  • Erwan Faou and Frédéric Rousset, Landau damping in Sobolev spaces for the Vlasov-HMF model, Arch. Ration. Mech. Anal. 219 (2016), no. 2, 887–902. MR 3437866, DOI 10.1007/s00205-015-0911-9
  • Bastien Fernandez, David Gérard-Varet, and Giambattista Giacomin, Landau damping in the Kuramoto model, Ann. Henri Poincaré 17 (2016), no. 7, 1793–1823. MR 3510470, DOI 10.1007/s00023-015-0450-9
  • Ragnar Fjørtoft, Application of integral theorems in deriving criteria of stability for laminar flows and for the baroclinic circular vortex, Geofys. Publ. Norske Vid.-Akad. Oslo 17 (1950), no. 6, 52. MR 53717
  • Susan Friedlander, Nataša Pavlović, and Roman Shvydkoy, Nonlinear instability for the Navier-Stokes equations, Comm. Math. Phys. 264 (2006), no. 2, 335–347. MR 2215608, DOI 10.1007/s00220-006-1526-7
  • Susan Friedlander, Walter Strauss, and Misha Vishik, Nonlinear instability in an ideal fluid, Ann. Inst. H. Poincaré C Anal. Non Linéaire 14 (1997), no. 2, 187–209 (English, with English and French summaries). MR 1441392, DOI 10.1016/S0294-1449(97)80144-8
  • Susan Friedlander and Misha M. Vishik, Instability criteria for the flow of an inviscid incompressible fluid, Phys. Rev. Lett. 66 (1991), no. 17, 2204–2206. MR 1102381, DOI 10.1103/PhysRevLett.66.2204
  • Isabelle Gallagher, Thierry Gallay, and Francis Nier, Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator, Int. Math. Res. Not. IMRN 12 (2009), 2147–2199. MR 2511908, DOI 10.1093/imrn/rnp013
  • Thierry Gallay, Enhanced dissipation and axisymmetrization of two-dimensional viscous vortices, Arch. Ration. Mech. Anal. 230 (2018), no. 3, 939–975. MR 3851053, DOI 10.1007/s00205-018-1262-0
  • T. Gebhardt and S. Grossmann, Chaos transition despite linear stability, Phys. Rev. E, 50 (1994), no. 5, 3705.
  • Andrew D. Gilbert, Spiral structures and spectra in two-dimensional turbulence, J. Fluid Mech. 193 (1988), 475–497. MR 985193, DOI 10.1017/S0022112088002228
  • Robert Glassey and Jack Schaeffer, Time decay for solutions to the linearized Vlasov equation, Transport Theory Statist. Phys. 23 (1994), no. 4, 411–453. MR 1264846, DOI 10.1080/00411459408203873
  • Robert Glassey and Jack Schaeffer, On time decay rates in Landau damping, Comm. Partial Differential Equations 20 (1995), no. 3-4, 647–676. MR 1318084, DOI 10.1080/03605309508821107
  • François Golse, Pierre-Louis Lions, Benoît Perthame, and Rémi Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal. 76 (1988), no. 1, 110–125. MR 923047, DOI 10.1016/0022-1236(88)90051-1
  • François Golse, Benoît Perthame, and Rémi Sentis, Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale d’un opérateur de transport, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 7, 341–344 (French, with English summary). MR 808622
  • Emmanuel Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math. 53 (2000), no. 9, 1067–1091. MR 1761409, DOI 10.1002/1097-0312(200009)53:9<1067::AID-CPA1>3.3.CO;2-H
  • Emmanuel Grenier, Yan Guo, and Toan T. Nguyen, Spectral instability of general symmetric shear flows in a two-dimensional channel, Adv. Math. 292 (2016), 52–110. MR 3464020, DOI 10.1016/j.aim.2016.01.007
  • Yan Guo, Variational method for stable polytropic galaxies, Arch. Ration. Mech. Anal. 150 (1999), no. 3, 209–224. MR 1738118, DOI 10.1007/s002050050187
  • Yan Guo and Gerhard Rein, Isotropic steady states in galactic dynamics, Comm. Math. Phys. 219 (2001), no. 3, 607–629. MR 1838751, DOI 10.1007/s002200100434
  • W. Heisenberg, On the stability of laminar flow, in Scientific Review Papers, Talks, and Books, Wissenschaftliche Übersichtsartikel, Vorträge und Bücher, pp. 471–475. Springer, 1984.
  • Darryl D. Holm, Jerrold E. Marsden, Tudor Ratiu, and Alan Weinstein, Nonlinear stability of fluid and plasma equilibria, Phys. Rep. 123 (1985), no. 1-2, 116. MR 794110, DOI 10.1016/0370-1573(85)90028-6
  • Hyung Ju Hwang and Juan J. L. Velázquez, On the existence of exponentially decreasing solutions of the nonlinear Landau damping problem, Indiana Univ. Math. J. 58 (2009), no. 6, 2623–2660. MR 2603762, DOI 10.1512/iumj.2009.58.3835
  • S. Ibrahim, Y. Maekawa, and N. Masmoudi, On pseudospectral bound for nonself-adjoint operators and its application to stability of kolmogorov flows, arXiv:1710.05132 (2017).
  • A. Ionescu and H. Jia, Inviscid damping near shear flows in a channel, arXiv:1808.04026 (2018).
  • Lord Kelvin, Stability of fluid motion—rectilinear motion of viscous fluid between two parallel plates, Phil. Mag. 24 (1887), 188.
  • Alexander Kiselev and Vladimir Šverák, Small scale creation for solutions of the incompressible two-dimensional Euler equation, Ann. of Math. (2) 180 (2014), no. 3, 1205–1220. MR 3245016, DOI 10.4007/annals.2014.180.3.9
  • P. Klebanoff, K. Tidstrom, and L. Sargent, The three-dimensional nature of boundary-layer instability, Journal of Fluid Mechanics, 12 (1962), no. 1, 1–34.
  • Gunilla Kreiss, Anders Lundbladh, and Dan S. Henningson, Bounds for the threshold amplitudes in subcritical shear flows, J. Fluid Mech. 270 (1994), 175–198. MR 1287784, DOI 10.1017/S0022112094004234
  • M. T. Landahl, A note on an algebraic instability of inviscid parallel shear flows, J. Fluid Mech. 98 (1980), no. 2, 243–251. MR 576172, DOI 10.1017/S0022112080000122
  • L. Landau, On the vibrations of the electronic plasma, Acad. Sci. USSR. J. Phys. 10 (1946), 25–34. MR 0023765
  • M. Latini and A. Bernoff, Transient anomalous diffusion in Poiseuille flow, Journal of Fluid Mechanics, 441 (2001), 399–411.
  • T. Li, D. Wei, and Z. Zhang, Pseudospectral and spectral bounds for the oseen vortices operator, arXiv:1701.06269 (2017).
  • T. Li, D. Wei, and Z. Zhang, Pseudospectral bound and transition threshold for the 3d Kolmogorov flow, arXiv:1801.05645 (2018).
  • Mattias Liefvendahl and Gunilla Kreiss, Bounds for the threshold amplitude for plane Couette flow, J. Nonlinear Math. Phys. 9 (2002), no. 3, 311–324. MR 1916388, DOI 10.2991/jnmp.2002.9.3.5
  • C. C. Lin, On the stability of two-dimensional parallel flows, Proc. Nat. Acad. Sci. U.S.A. 30 (1944), 316–323. MR 11614, DOI 10.1073/pnas.30.10.316
  • Z. Lin and M. Xu, Metastability of Kolmogorov flows and inviscid damping of shear flows, arXiv:1707.00278 (2017).
  • Zhiwu Lin and Chongchun Zeng, Inviscid dynamical structures near Couette flow, Arch. Ration. Mech. Anal. 200 (2011), no. 3, 1075–1097. MR 2796139, DOI 10.1007/s00205-010-0384-9
  • R. Lindzen, Instability of plane parallel shear flow (toward a mechanistic picture of how it works), PAGEOPH 126 (1988), 1.
  • A. Lundbladh, D. S. Henningson, and S. C. Reddy, Threshold amplitudes for transition in channel flows, in Transition, turbulence and combustion, pp. 309–318. Springer, 1994.
  • T. Lundgren, Strained spiral vortex model for turbulent fine structure, Phys. of Fl. 25 (1982), 2193.
  • D. Lynden-Bell, Statistical mechanics of violent relaxation in stellar systems, Mon. Not. R. Astr. Soc. 136 (1967), 101–121.
  • Andrew J. Majda and Andrea L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, vol. 27, Cambridge University Press, Cambridge, 2002. MR 1867882
  • J. Malmberg, C. Wharton, C. Gould, and T. O’Neil, Plasma wave echo, Phys. Rev. Lett. 20 (1968), no. 3, 95–97.
  • Carlo Marchioro and Mario Pulvirenti, Mathematical theory of incompressible nonviscous fluids, Applied Mathematical Sciences, vol. 96, Springer-Verlag, New York, 1994. MR 1245492, DOI 10.1007/978-1-4612-4284-0
  • J. D. McCalpin, On the adjustment of azimuthally perturbed vortices, Journal of Geophysical Research: Oceans, 92 (1987), C8, 8213–8225.
  • M. Melander, J. McWilliams, and N. Zabusky, Axisymmetrization and vorticity-gradient intensification of an isolated two-dimensional vortex through filamentation, Journal of Fluid Mechanics 178 (1987), 137–159.
  • P. J. Morrison, Hamiltonian description of the ideal fluid, Rev. Modern Phys. 70 (1998), no. 2, 467–521. MR 1627532, DOI 10.1103/RevModPhys.70.467
  • Clément Mouhot and Cédric Villani, On Landau damping, Acta Math. 207 (2011), no. 1, 29–201. MR 2863910, DOI 10.1007/s11511-011-0068-9
  • N. S. Nadirashvili, Wandering solutions of the two-dimensional Euler equation, Funktsional. Anal. i Prilozhen. 25 (1991), no. 3, 70–71 (Russian); English transl., Funct. Anal. Appl. 25 (1991), no. 3, 220–221 (1992). MR 1139875, DOI 10.1007/BF01085491
  • W. Orr, The stability or instability of steady motions of a perfect liquid and of a viscous liquid, Part I: a perfect liquid, Proc. Royal Irish Acad. Sec. A: Math. Phys. Sci. 27 (1907), 9–68.
  • S. A. Orszag and L. C. Kells, Transition to turbulence in plane Poiseuille and plane Couette flow, Journal of Fluid Mechanics, 96 (1980), no. 1, 159–205.
  • O. Penrose, Electrostatic instability of a uniform nonMaxwellian plasma, Phys. Fluids 3 (1960), 258–265.
  • D. S. Pradeep and F. Hussain, Transient growth of perturbations in a vortex column, J. Fluid Mech. 550 (2006), 251–288. MR 2263985, DOI 10.1017/S0022112005008207
  • Lord Rayleigh, On the Stability, or Instability, of certain Fluid Motions, Proc. Lond. Math. Soc. 11 (1879/80), 57–70. MR 1575266, DOI 10.1112/plms/s1-11.1.57
  • L. Rayleigh. On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 32 (1916), no. 192, 529–546.
  • Satish C. Reddy, Peter J. Schmid, Jeffrey S. Baggett, and Dan S. Henningson, On stability of streamwise streaks and transition thresholds in plane channel flows, J. Fluid Mech. 365 (1998), 269–303. MR 1631950, DOI 10.1017/S0022112098001323
  • 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. 174 (1883), 935–982.
  • P. Rhines and W. Young, How rapidly is a passive scalar mixed within closed streamlines?, Journal of Fluid Mechanics 133 (1983), 133–145.
  • V. A. Romanov, Stability of plane-parallel Couette flow, Funkcional. Anal. i Priložen. 7 (1973), no. 2, 62–73 (Russian). MR 0326191
  • D. Ryutov, Landau damping: half a century with the great discovery, Plasma Physics and Controlled Fusion 41 (1999), no. 3A.
  • D. A. Schecter, D. H. E. Dubin, A. C. Cass, C. F. Driscoll, I. M. Lansky, and T. M. O’Neil, Inviscid damping of asymmetries on a two-dimensional vortex, Phys. Fluids 12 (2000), no. 10, 2397–2412. MR 1789996, DOI 10.1063/1.1289505
  • Peter J. Schmid and Dan S. Henningson, Stability and transition in shear flows, Applied Mathematical Sciences, vol. 142, Springer-Verlag, New York, 2001. MR 1801992, DOI 10.1007/978-1-4613-0185-1
  • R. Shvydkoy and S. Friedlander, Problem for the linearized Euler equation, Nonlinear Partial Differential Equations and Related Analysis: The Emphasis Year 2002-2003 Program on Nonlinear Partial Differential Equations and Related Analysis, September 2002-July 2003, Northwestern University, Evanston, Illinois, vol. 371, p. 271, 2005.
  • G. B. Smith and M. T. Montgomery, Vortex axisymmetrization: Dependence on azimuthal wave-number or asymmetric radial structure changes, Quarterly Journal of the Royal Meteorological Society 121 (1995), no. 527, 1615–1650.
  • H. Squire, On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 142 (1933), no. 847, 621–628.
  • Terence Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. MR 2233925, DOI 10.1090/cbms/106
  • N. Tillmark and P. Alfredsson, Experiments on transition in plane Couette flow, J. Fluid Mech. 235 (1992), 89–102.
  • Lloyd N. Trefethen and Mark Embree, Spectra and pseudospectra, Princeton University Press, Princeton, NJ, 2005. The behavior of nonnormal matrices and operators. MR 2155029
  • Lloyd N. Trefethen, Anne E. Trefethen, Satish C. Reddy, and Tobin A. Driscoll, Hydrodynamic stability without eigenvalues, Science 261 (1993), no. 5121, 578–584. MR 1229495, DOI 10.1126/science.261.5121.578
  • N. G. van Kampen, On the theory of stationary waves in plasmas, Physica 21 (1955), 949–963. MR 75080
  • J. Vanneste, Nonlinear dynamics of anisotropic disturbances in plane Couette flow, SIAM J. Appl. Math. 62 (2001/02), no. 3, 924–944. MR 1897729, DOI 10.1137/S0036139900381420
  • J. Vanneste, P. Morrison, and T. Warn, Strong echo effect and nonlinear transient growth in shear flows, Physics of Fluids, 10 (1998), no. 6, 1398.
  • J. Vukadinovic, E. Dedits, A. C. Poje, and T. Schäfer, Averaging and spectral properties for the 2D advection-diffusion equation in the semi-classical limit for vanishing diffusivity, Phys. D 310 (2015), 1–18. MR 3396847, DOI 10.1016/j.physd.2015.07.011
  • Fabian Waleffe, Transition in shear flows. Nonlinear normality versus non-normal linearity, Phys. Fluids 7 (1995), no. 12, 3060–3066. MR 1361366, DOI 10.1063/1.868682
  • Y. H. Wan and M. Pulvirenti, Nonlinear stability of circular vortex patches, Comm. Math. Phys. 99 (1985), no. 3, 435–450. MR 795112
  • Y. Wang, Rapid filamentation zone in a numerically simulated tropical cyclone, Journal of the Atmospheric Sciences 65 (2008), no. 4, 1158–1181.
  • D. Wei and Z. Zhang, Transition threshold for the 3D Couette flow in Sobolev space, arXiv:1803.01359 (2018).
  • Dongyi Wei, Zhifei Zhang, and Weiren Zhao, Linear inviscid damping for a class of monotone shear flow in Sobolev spaces, Comm. Pure Appl. Math. 71 (2018), no. 4, 617–687. MR 3772399, DOI 10.1002/cpa.21672
  • D. Wei, Z. Zhang, and W. Zhao, Linear inviscid damping and enhanced dissipation for the Kolmogorov flow, arXiv:1711.01822 (2017).
  • D. Wei, Z. Zhang, and W. Zhao, Linear inviscid damping and vorticity depletion for shear flows, arXiv:1704.00428 (2017).
  • Akiva M. Yaglom, Hydrodynamic instability and transition to turbulence, Fluid Mechanics and its Applications, vol. 100, Springer, Dordrecht, 2012. With a foreword by Uriel Frisch and a memorial note for Yaglom by Peter Bradshaw. MR 3185102, DOI 10.1007/978-94-007-4237-6
  • Jincheng Yang and Zhiwu Lin, Linear inviscid damping for Couette flow in stratified fluid, J. Math. Fluid Mech. 20 (2018), no. 2, 445–472. MR 3808578, DOI 10.1007/s00021-017-0328-3
  • H. Yao and N. Zabusky, Axisymmetrization of an isolated vortex region by splitting and partial merging of satellite depletion perturbations, Physics of Fluids 8 (1996), no. 7, 1842–1847.
  • Brent Young, Landau damping in relativistic plasmas, J. Math. Phys. 57 (2016), no. 2, 021502, 68. MR 3448678, DOI 10.1063/1.4939275
  • J. Yu and C. Driscoll, Diocotron wave echoes in a pure electron plasma, IEEE Trans. Plasma Sci. 30 (2005), no. 1, 24–25.
  • J. Yu, C. Driscoll, and T. O‘Neil, Phase mixing and echoes in a pure electron plasma, Phys. of Plasmas 12 (2005), 055701.
  • Christian Zillinger, Linear inviscid damping for monotone shear flows, Trans. Amer. Math. Soc. 369 (2017), no. 12, 8799–8855. MR 3710645, DOI 10.1090/tran/6942
  • Christian Zillinger, 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 (2016), no. 3, 1449–1509. MR 3509006, DOI 10.1007/s00205-016-0991-1
  • Christian Zillinger, 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 (2016), no. 3, 1449–1509. MR 3509006, DOI 10.1007/s00205-016-0991-1
  • Christian Zillinger, On circular flows: linear stability and damping, J. Differential Equations 263 (2017), no. 11, 7856–7899. MR 3705700, DOI 10.1016/j.jde.2017.08.026
  • Christian Zillinger, Linear inviscid damping for monotone shear flows, Trans. Amer. Math. Soc. 369 (2017), no. 12, 8799–8855. MR 3710645, DOI 10.1090/tran/6942
  • Andrej Zlatoš, Diffusion in fluid flow: dissipation enhancement by flows in 2D, Comm. Partial Differential Equations 35 (2010), no. 3, 496–534. MR 2748635, DOI 10.1080/03605300903362546
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Additional Information
  • Jacob Bedrossian
  • Affiliation: 4176 Campus Drive - William E. Kirwan Hall, College Park, Maryland 20742
  • MR Author ID: 908903
  • Email: jacob@cscamm.umd.edu
  • Pierre Germain
  • Affiliation: Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012
  • MR Author ID: 758713
  • Email: pgermain@cims.nyu.edu
  • Nader Masmoudi
  • Affiliation: Department of mathematics, New York University in Abu Dhabi, Saadyiat Island, Abu Dhabi, United Arab Emirates
  • MR Author ID: 620387
  • Email: masmoudi@cims.nyu.edu
  • Received by editor(s): March 6, 2018
  • Published electronically: November 15, 2018
  • Additional Notes: The first author was supported by NSF DMS-1552826.
    The second author was supported by NSF DMS-1501019.
    The third author was supported by DMS-1716466.
  • © Copyright 2018 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 56 (2019), 373-414
  • MSC (2010): Primary 76-02, 35-02, 76E05, 76E30, 35B25, 35B35, 35B34
  • DOI: https://doi.org/10.1090/bull/1649
  • MathSciNet review: 3974608