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Dynamics Near the Subcritical Transition of the 3D Couette Flow I: Below Threshold Case

About this Title

Jacob Bedrossian, Pierre Germain and Nader Masmoudi

Publication: Memoirs of the American Mathematical Society
Publication Year: 2020; Volume 266, Number 1294
ISBNs: 978-1-4704-4217-0 (print); 978-1-4704-6251-2 (online)
DOI: https://doi.org/10.1090/memo/1294
Published electronically: July 21, 2020
MSC: Primary 35B35, 76E05, 76E30, 76F06, 76F10; Secondary 35B40, 76F25

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Outline of the proof
  • 3. Regularization and continuation
  • 4. High norm estimate on $Q^2$
  • 5. High norm estimate on $Q^3$
  • 6. High norm estimate on $Q^1_0$
  • 7. High norm estimate on $Q^1_{\neq }$
  • 8. Coordinate system controls
  • 9. Enhanced dissipation estimates
  • 10. Sobolev estimates
  • Acknowledgments
  • A. Fourier analysis conventions, elementary inequalities, and Gevrey spaces
  • B. Definition and analysis of norms
  • C. Multiplier and paraproduct tools
  • D. Elliptic estimates

Abstract

We study small disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number Re. We prove that for sufficiently regular initial data of size $\epsilon \leq c_0\mathbf {Re}^{-1}$ for some universal $c_0 > 0$, the solution is global, remains within $O(c_0)$ of the Couette flow in $L^2$, and returns to the Couette flow as $t \rightarrow \infty$. For times $t \gtrsim \mathbf {Re}^{1/3}$, the streamwise dependence is damped by a mixing-enhanced dissipation effect and the solution is rapidly attracted to the class of “2.5 dimensional” streamwise-independent solutions referred to as streaks. Our analysis contains perturbations that experience a transient growth of kinetic energy from $O(\mathbf {Re}^{-1})$ to $O(c_0)$ due to the algebraic linear instability known as the lift-up effect. Furthermore, solutions can exhibit a direct cascade of energy to small scales. The behavior is very different from the 2D Couette flow, in which stability is independent of $\mathbf {Re}$, enstrophy experiences a direct cascade, and inviscid damping is dominant (resulting in a kind of inverse energy cascade). In 3D, inviscid damping will play a role on one component of the velocity, but the primary stability mechanism is the mixing-enhanced dissipation. Central to the proof is a detailed analysis of the interplay between the stabilizing effects of the mixing and enhanced dissipation and the destabilizing effects of the lift-up effect, vortex stretching, and weakly nonlinear instabilities connected to the non-normal nature of the linearization.

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References
  • 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
  • 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
  • Hajer Bahouri, Jean-Yves Chemin, and Raphaël Danchin, Fourier analysis and nonlinear partial differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 343, Springer, Heidelberg, 2011. MR 2768550
  • Konrad Bajer, Andrew P. Bassom, and Andrew D. Gilbert, Accelerated diffusion in the centre of a vortex, J. Fluid Mech. 437 (2001), 395–411. MR 1841799, DOI 10.1017/S0022112001004232
  • N.J. Balmforth and P.J. Morrison. Normal modes and continuous spectra. Annals of the New York Academy of Sciences, 773(1):80–94, 1995.
  • N.J. Balmforth, P.J. Morrison, and J.-L. Thiffeault. Pattern formation in Hamiltonian systems with continuous spectra; a normal-form single-wave model. preprint, 2013.
  • 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
  • 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, 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
  • 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
  • Jacob Bedrossian and Nader Masmoudi. Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations. Publications mathématiques de l’IHÉS, pages 1–106, 2013.
  • A.J. Bernoff and J.F. Lingevitch. Rapid relaxation of an axisymmetric vortex. Phys. Fluids, 6(3717), 1994.
  • 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. Phys. of Fluids, 10:2597, 1998.
  • Freddy Bouchet and Hidetoshi Morita, Large time behavior and asymptotic stability of the 2D Euler and linearized Euler equations, Phys. D 239 (2010), no. 12, 948–966. MR 2639613, DOI 10.1016/j.physd.2010.01.020
  • R.J. Briggs, J.D. Daugherty, and R.H. Levy. Role of Landau damping in crossed-field electron beams and inviscid shear flow. Phys. Fl., 13(2), 1970.
  • 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
  • A.J. Cerfon, J.P. Freidberg, F.I. Parra, and T.A. Antaya. Analytic fluid theory of beam spiraling in high-intensity cyclotrons. Phys. Rev. ST Accel. Beams, 16(024202), 2013.
  • S. Jonathan Chapman, Subcritical transition in channel flows, J. Fluid Mech. 451 (2002), 35–97. MR 1886008, DOI 10.1017/S0022112001006255
  • Demetrios Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math. 39 (1986), no. 2, 267–282. MR 820070, DOI 10.1002/cpa.3160390205
  • 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
  • Alex DD Craik. Non-linear resonant instability in boundary layers. Journal of Fluid Mechanics, 50(02):393–413, 1971.
  • F Daviaud, J Hegseth, and P Bergé. Subcritical transition to turbulence in plane Couette flow. Phys. rev. lett., 69(17):2511, 1992.
  • 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
  • B Dubrulle and S Nazarenko. On scaling laws for the transition to turbulence in uniform-shear flows. Euro. Phys. Lett., 27(2):129, 1994.
  • 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., 650:119–129, 2010.
  • T Ellingsen and E Palm. Stability of linear flow. Phys. of Fluids, 18:487, 1975.
  • Per A Elofsson, Mitsuyoshi Kawakami, and P Henrik Alfredsson. Experiments on the stability of streamwise streaks in plane Poiseuille flow. Physics of Fluids (1994–present), 11(4):915–930, 1999.
  • 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
  • C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal. 87 (1989), no. 2, 359–369. MR 1026858, DOI 10.1016/0022-1236(89)90015-3
  • T. Gebhardt and S. Grossmann. Chaos transition despite linear stability. Phys. Rev. E, 50(5):3705, 1994.
  • Maurice Gevrey, Sur la nature analytique des solutions des équations aux dérivées partielles. Premier mémoire, Ann. Sci. École Norm. Sup. (3) 35 (1918), 129–190 (French). MR 1509208
  • 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
  • Emmanuel Grenier, Yan Guo, and Toan T. Nguyen, Spectral instability of characteristic boundary layer flows, Duke Math. J. 165 (2016), no. 16, 3085–3146. MR 3566199, DOI 10.1215/00127094-3645437
  • Dan S Henningson, Anders Lundbladh, and Arne V Johansson. A mechanism for bypass transition from localized disturbances in wall-bounded shear flows. J. of Fluid Mech., 250:169–207, 1993.
  • B Hof, A Juel, and T Mullin. Scaling of the turbulence transition threshold in a pipe. Phys. rev. let., 91(24):244502, 2003.
  • Lars Hörmander, The Nash-Moser theorem and paradifferential operators, Analysis, et cetera, Academic Press, Boston, MA, 1990, pp. 429–449. MR 1039355
  • 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
  • Lord Kelvin. Stability of fluid motion-rectilinear motion of viscous fluid between two parallel plates. Phil. Mag., (24):188, 1887.
  • S. Klainerman, The null condition and global existence to nonlinear wave equations, Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984) Lectures in Appl. Math., vol. 23, Amer. Math. Soc., Providence, RI, 1986, pp. 293–326. MR 837683
  • PS Klebanoff, KD Tidstrom, and LM Sargent. The three-dimensional nature of boundary-layer instability. Journal of Fluid Mechanics, 12(01):1–34, 1962.
  • 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
  • Igor Kukavica and Vlad Vicol, On the radius of analyticity of solutions to the three-dimensional Euler equations, Proc. Amer. Math. Soc. 137 (2009), no. 2, 669–677. MR 2448589, DOI 10.1090/S0002-9939-08-09693-7
  • 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.J. Bernoff. Transient anomalous diffusion in Poiseuille flow. J. of Fluid Mech., 441:399–411, 2001.
  • G. Lemoult, J.-L. Aider, and J.E. Wesfreid. Experimental scaling law for the subcritical transition to turbulence in plane Poiseuille flow. Phy. Rev. E, 85(2):025303, 2012.
  • C. David Levermore and Marcel Oliver, Analyticity of solutions for a generalized Euler equation, J. Differential Equations 133 (1997), no. 2, 321–339. MR 1427856, DOI 10.1006/jdeq.1996.3200
  • Y. Charles Li and Zhiwu Lin, A resolution of the Sommerfeld paradox, SIAM J. Math. Anal. 43 (2011), no. 4, 1923–1954. MR 2831254, DOI 10.1137/100794912
  • 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
  • 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
  • Anders Lundbladh, Dan S Henningson, and Satish C Reddy. Threshold amplitudes for transition in channel flows. In Transition, turbulence and combustion, pages 309–318. Springer, 1994.
  • T.S. Lundgren. Strained spiral vortex model for turbulent fine structure. Phys. of Fl., 25:2193, 1982.
  • 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(3):95–97, 1968.
  • M. T. Montgomery and R. J. Kallenbach. A theory for vortex Rossby-waves and its application to spiral bands and intensity changes in hurricanes. Quarterly Journal of the Royal Meteorological Society, 123(538):435–465, 1997.
  • 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
  • T. Mullin, Experimental studies of transition to turbulence in a pipe, Annual review of fluid mechanics. Volume 43, 2011, Annu. Rev. Fluid Mech., vol. 43, Annual Reviews, Palo Alto, CA, 2011, pp. 1–24. MR 2768009, DOI 10.1146/annurev-fluid-122109-160652
  • L. Nirenberg, An abstract form of the nonlinear Cauchy-Kowalewski theorem, J. Differential Geometry 6 (1972), 561–576. MR 322321
  • Takaaki Nishida, A note on a theorem of Nirenberg, J. Differential Geometry 12 (1977), no. 4, 629–633 (1978). MR 512931
  • M Nishioka, Y Ichikawa, et al. An experimental investigation of the stability of plane Poiseuille flow. Journal of Fluid Mechanics, 72(04):731–751, 1975.
  • 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., 27:9–68, 1907.
  • Steven A Orszag and Lawrence C Kells. Transition to turbulence in plane Poiseuille and plane Couette flow. J. of Fluid Mech., 96(1):159–205, 1980.
  • 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
  • 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., (35):84, 1883.
  • P.B. Rhines and W.R. Young. How rapidly is a passive scalar mixed within closed streamlines? J. of Fluid Mech., 133:133–145, 1983.
  • V. A. Romanov, Stability of plane-parallel Couette flow, Funkcional. Anal. i Priložen. 7 (1973), no. 2, 62–73 (Russian). MR 0326191
  • D.D. Ryutov. Landau damping: half a century with the great discovery. Plasma physics and controlled fusion, 41(3A):A1, 1999.
  • 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
  • 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(527):1615–1650, 1995.
  • N. Tillmark and P.H. Alfredsson. Experiments on transition in plane Couette flow. J. Fluid Mech., 235:89–102, 1992.
  • 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
  • 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.J. Morrison, and T. Warn. Strong echo effect and nonlinear transient growth in shear flows. Physics of Fluids, 10:1398, 1998.
  • 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
  • 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
  • Brent Young, Landau damping in relativistic plasmas, J. Math. Phys. 57 (2016), no. 2, 021502, 68. MR 3448678, DOI 10.1063/1.4939275
  • J.H. Yu and C.F. Driscoll. Diocotron wave echoes in a pure electron plasma. IEEE Trans. Plasma Sci., 30(1), 2002.
  • J.H. Yu, C.F. Driscoll, and T.M. O‘Neil. Phase mixing and echoes in a pure electron plasma. Phys. of Plasmas, 12(055701), 2005.
  • 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