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On Landau damping

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Acta Mathematica

Abstract

Going beyond the linearized study has been a longstanding problem in the theory of Landau damping. In this paper we establish exponential Landau damping in analytic regularity. The damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables, rather than exchanges of energy; phase mixing is the driving mechanism. The analysis involves new families of analytic norms, measuring regularity by comparison with solutions of the free transport equation; new functional inequalities; a control of non-linear echoes; sharp “deflection” estimates; and a Newton approximation scheme. Our results hold for any potential no more singular than Coulomb or Newton interaction; the limit cases are included with specific technical effort. As a side result, the stability of homogeneous equilibria of the non-linear Vlasov equation is established under sharp assumptions. We point out the strong analogy with the KAM theory, and discuss physical implications. Finally, we extend these results to some Gevrey (non-analytic) distribution functions.

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References

  1. Akhiezer, A., Akhiezer, I., Polovin, R., Sitenko, A. & Stepanov, K., Plasma Electrodynamics. Vol. I: Linear Theory. Vol. II: Non-Linear Theory and Fluctuations. Pergamon Press, Oxford–New York, 1975.

    Google Scholar 

  2. Alinhac, S. & Gérard, P., Pseudo-Differential Operators and the Nash–Moser Theorem. Graduate Studies in Mathematics, 82. Amer. Math. Soc., Providence, RI, 2007.

  3. Bach, V., Fröhlich, J. & Sigal, I. M., Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field. Comm. Math. Phys., 207 (1999), 249–290.

    Article  MATH  MathSciNet  Google Scholar 

  4. Backus, G., Linearized plasma oscillations in arbitrary electron velocity distributions. J. Math. Phys., 1 (1960), 178–191; erratum, 559.

    Article  MATH  MathSciNet  Google Scholar 

  5. Balescu, R., Statistical Mechanics of Charged Particles. Monographs in Statistical Physics and Thermodynamics, 4. Wiley, London–New York–Sydney, 1963.

    Google Scholar 

  6. Batt, J. & Rein, G., Global classical solutions of the periodic Vlasov–Poisson system in three dimensions. C. R. Acad. Sci. Paris Sér. I Math., 313 (1991), 411–416.

    MATH  MathSciNet  Google Scholar 

  7. Belmont, G., Mottez, F., Chust, T. & Hess, S., Existence of non-Landau solutions for Langmuir waves. Phys. of Plasmas, 15 (2008), 052310, 1–14.

    Google Scholar 

  8. Benachour, S., Analyticité des solutions des équations de Vlasov–Poisson. Ann. Sc. Norm. Super. Pisa Cl. Sci., 16 (1989), 83–104.

    MATH  MathSciNet  Google Scholar 

  9. Binney, J. & Tremaine, S., Galactic Dynamics. First edition. Princeton Series in Astrophysics. Princeton University Press, Princeton, 1987.

    MATH  Google Scholar 

  10. — Galactic Dynamics. Second edition. Princeton Series in Astrophysics. Princeton University Press, Princeton, 2008.

  11. Bouchet, F., Stochastic process of equilibrium fluctuations of a system with long-range interactions. Phys. Rev. E, 70 (2004), 036113, 1–4.

    Google Scholar 

  12. Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal., 3 (1993), 107–156.

    Article  MATH  MathSciNet  Google Scholar 

  13. Caglioti, E. & MaffeI, C., Time asymptotics for solutions of Vlasov–Poisson equation in a circle. J. Stat. Phys., 92 (1998), 301–323.

    Article  MATH  MathSciNet  Google Scholar 

  14. CasE, K. M., Plasma oscillations. Ann. Physics, 7 (1959), 349–364.

    Article  MATH  MathSciNet  Google Scholar 

  15. Chavanis, P. H., Quasilinear theory of the 2D Euler equation. Phys. Rev. Lett., 84:24 (2000), 5512–5515.

    Article  MathSciNet  Google Scholar 

  16. Chavanis, P. H., Sommeria, J., & Robert, R., Statistical mechanics of two-dimensional vortices and collisionless stellar systems. Astrophys. J., 471 (1996), 385–399.

    Article  Google Scholar 

  17. Chemin, J. Y., 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., 9, pp. 99–123. Soc. Math. France, Paris, 2004.

  18. Chierchia, L., A. N. Kolmogorov’s 1954 paper on nearly-integrable Hamiltonian systems. A comment on: “On conservation of conditionally periodic motions for a small change in Hamilton’s function” [Dokl. Akad. Nauk SSSR, 98 (1954), 527–530]. Regul. Chaotic Dyn., 13 (2008), 130–139.

  19. Chust, T., Belmont, G., Mottez, F. & Hess, S., Landau and non-Landau linear damping: Physics of the dissipation. Phys. Plasmas, 16 (2009), 092104, 13 pp.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. Dereziński, J. & Gérard, C., Scattering Theory of Classical and Quantum N-Particle Systems. Texts and Monographs in Physics. Springer, Berlin–Heidelberg, 1997.

    Google Scholar 

  22. Desvillettes, L. & Villani, C., On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker–Planck equation. Comm. Pure Appl. Math., 54 (2001), 1–42.

    Article  MATH  MathSciNet  Google Scholar 

  23. Elskens, Y., Irreversible behaviours in Vlasov equation and many-body Hamiltonian dynamics: Landau damping, chaos and granularity, in Topics in Kinetic Theory, Fields Inst. Commun., 46, pp. 89–108. Amer. Math. Soc., Providence, RI, 2005.

  24. Elskens, Y. & Escande, D. F., Microscopic Dynamics of Plasmas and Chaos. Institute of Physics, Bristol, 2003.

    Book  MATH  Google Scholar 

  25. Escande, D. F., Wave-particle interaction in plasmas: a qualitative approach, in Long-Range Interacting Systems, pp. 13–14, 469–506. Oxford University Press, Oxford, 2010.

  26. Fathi, A., Weak KAM Theory in Lagrangian Dynamics. Cambridge University Press, Cambridge, 2010.

    Google Scholar 

  27. Filbet, F., Numerical simulations. Available online at http://math.univ-lyon1.fr/~filbet/publication.html.

  28. Fridman, A. M. & Polyachenko, V. L., Physics of Gravitating Systems. Vol. I: Equilibrium and Stability. Vol. II: Nonlinear Collective Processes. Astrophysical Applications. Springer, New York, 1984.

    Google Scholar 

  29. Gangbo, W. & Tudorascu, A., Lagrangian dynamics on an infinite-dimensional torus; a weak KAM theorem. Adv. Math., 224 (2010), 260–292.

    Article  MATH  MathSciNet  Google Scholar 

  30. Glassey, R. & Schaeffer, J., Time decay for solutions to the linearized Vlasov equation. Transport Theory Statist. Phys., 23 (1994), 411–453.

    Article  MATH  MathSciNet  Google Scholar 

  31. — On time decay rates in Landau damping. Comm. Partial Differential Equations, 20 (1995), 647–676.

  32. Gould, R., O’Neil, T. & Malmberg, J., Plasma wave echo. Phys. Rev. Letters, 19:5 (1967), 219–222.

    Article  Google Scholar 

  33. Gross, L., Logarithmic Sobolev inequalities. Amer. J. Math., 97 (1975), 1061–1083.

    Article  MathSciNet  Google Scholar 

  34. Guo, Y. & Rein, G., A non-variational approach to non-linear stability in stellar dynamics applied to the King model. Comm. Math. Phys., 271 (2007), 489–509.

    Article  MATH  MathSciNet  Google Scholar 

  35. Guo, Y. & Strauss, W. A., Nonlinear instability of double-humped equilibria. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 12 (1995), 339–352.

    MATH  MathSciNet  Google Scholar 

  36. ter Haar, D., Men of Physics: L. D. Landau. Selected Reading of Physics, 2. Pergamon Press, Oxford–New York, 1969.

    Google Scholar 

  37. Hauray, M. & Jabin, P.E., N-particles approximation of the Vlasov equations with singular potential. Arch. Ration. Mech. Anal., 183 (2007), 489–524.

    Article  MATH  MathSciNet  Google Scholar 

  38. Hayes, J.N., On non-Landau damped solutions to the linearized Vlasov equation. Nuovo Cimento, 10:30 (1963), 1048–1063.

    MathSciNet  Google Scholar 

  39. Heath, R., Gamba, I., Morrison, P. & Michler, C., A discontinuous Galerkin method for the Vlasov–Poisson system. To appear in J. Comput. Phys.

  40. Horst, E., On the asymptotic growth of the solutions of the Vlasov–Poisson system. Math. Methods Appl. Sci., 16 (1993), 75–86.

    Article  MATH  MathSciNet  Google Scholar 

  41. Hwang, H. J. & Velázquez, J. J. L., On the existence of exponentially decreasing solutions of the nonlinear Landau damping problem. Indiana Univ. Math. J., 58:6 (2009), 2623–2660.

    Article  MATH  MathSciNet  Google Scholar 

  42. Isichenko, M., Nonlinear Landau damping in collisionless plasma and inviscid fluid. Phys. Rev. Lett., 78:12 (1997), 2369–2372.

    Article  Google Scholar 

  43. Jabin, P.E., Averaging lemmas and dispersion estimates for kinetic equations. Riv. Mat. Univ. Parma, 1 (2009), 71–138.

    MATH  MathSciNet  Google Scholar 

  44. Kaganovich, I.D., Effects of collisions and particle trapping on collisionless heating. Phys. Rev. Lett., 82:2 (1999), 327–330.

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  46. Kandrup, H., Violent relaxation, phase mixing, and gravitational Landau damping. Astrophys. J., 500 (1998), 120–128.

    Article  Google Scholar 

  47. Kiessling, M. K.-H., The “Jeans swindle”: a true story—mathematically speaking. Adv. in Appl. Math., 31 (2003), 132–149.

    Article  MATH  MathSciNet  Google Scholar 

  48. — Personal communication, 2009.

  49. Krall, N. & Trivelpiece, A., Principles of Plasma Physics. San Francisco Press, San Francisco, 1986.

    Google Scholar 

  50. Kuksin, S. B., Nearly Integrable Infinite-Dimensional Hamiltonian Systems. Lecture Notes in Mathematics, 1556. Springer, Berlin–Heidelberg, 1993.

    Google Scholar 

  51. — Analysis of Hamiltonian PDEs. Oxford Lecture Series in Mathematics and its Applications, 19. Oxford University Press, Oxford, 2000.

  52. Landau, L., On the vibrations of the electronic plasma. Akad. Nauk SSSR. Zhurnal Eksper. Teoret. Fiz., 16 (1946), 574–586 (Russian); English translation in Acad. Sci. USSR. J. Phys., 10 (1946), 25–34. Reproduced in [36].(19) There is a misprint in formula (17) of this reference (p. 104): replace e −(ka)2/2 by e −1/2(ka)2.

  53. Lemou, M., Méhats, F. & Raphaël, P., Orbital stability of spherical galactic models. To appear in Invent. Math.

  54. Lifshitz, E. M. & Pitaevskiĭ, L.P., Course of Theoretical Physics [“Landau–Lifshits”]. Vol. 10. Nauka, Moscow, 1979. English translation in Pergamon International Library of Science, Technology, Engineering and Social Studies. Pergamon Press, Oxford–New York, 1981.

    Google Scholar 

  55. Lin, Z. & Zeng, C., BGK waves and non-linear Landau damping. To appear in Comm. Math. Phys.

  56. Lions, P. L. & Perthame, B., Propagation of moments and regularity for the 3-dimensional Vlasov–Poisson system. Invent. Math., 105 (1991), 415–430.

    Article  MATH  MathSciNet  Google Scholar 

  57. Lynden-Bell, D., The stability and vibrations of a gas of stars. Mon. Not. R. Astr. Soc., 124 (1962), 279–296.

    MATH  Google Scholar 

  58. — Statistical mechanics of violent relaxation in stellar systems. Mon. Not. R. Astr. Soc., 136 (1967), 101–121.

  59. Malmberg, J. & Wharton, C., Collisionless damping of electrostatic plasma waves. Phys. Rev. Lett., 13:6 (1964), 184–186.

    Article  Google Scholar 

  60. Malmberg, J., Wharton, C., Gould, R. & O’Neil, T., Plasma wave echo experiment. Phys. Rev. Letters, 20:3 (1968), 95–97.

    Article  Google Scholar 

  61. Manfredi, G., Long-time behavior of non-linear Landau damping. Phys. Rev. Lett., 79:15 (1997), 2815–2818.

    Article  Google Scholar 

  62. Marchioro, C. & Pulvirenti, M., Mathematical Theory of Incompressible Nonviscous Fluids. Applied Mathematical Sciences, 96. Springer, New York, 1994.

    Book  Google Scholar 

  63. Maslov, V. P. & Fedoryuk, M.V., The linear theory of Landau damping. Mat. Sb., 127(169) (1985), 445–475, 559 (Russian); English translation in Math. USSR–Sb., 55 (1986), 437–465.

  64. Medvedev, M. V., Diamond, P. H., Rosenbluth, M. N. & Shevchenko, V. I., Asymptotic theory of non-linear Landau damping and particle trapping in waves of finite amplitude. Phys. Rev. Lett., 81:26 (1998), 5824–5827.

    Article  Google Scholar 

  65. Miller, J., Statistical mechanics of Euler equations in two dimensions. Phys. Rev. Lett., 65:17 (1990), 2137–2140.

    Article  MATH  MathSciNet  Google Scholar 

  66. Morrison, P. J., Hamiltonian description of Vlasov dynamics: action-angle variables for the continuous spectrum, in Proceedings of the Fifth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Maui, HI, 1998), Transport Theory Statist. Phys., 29, pp. 397–414. Taylor & Francis, Philadelphia, PA, 2000.

  67. Moser, J., A rapidly convergent iteration method and non-linear differential equations. II. Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (1966), 499–535.

    Google Scholar 

  68. — Recollections, in The Arnoldfest (Toronto, ON, 1997), Fields Inst. Commun., 24, pp. 19–21. Amer. Math. Soc., Providence, RI, 1999.

  69. Mouhot, C. & Villani, C., Landau damping. J. Math. Phys., 51 (2010), 015204, 7.

    MathSciNet  Google Scholar 

  70. Nash, J., The imbedding problem for Riemannian manifolds. Ann. of Math., 63 (1956), 20–63.

    Article  MATH  MathSciNet  Google Scholar 

  71. Nekhoroshev, N. N., An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. Uspekhi Mat. Nauk, 32 (1977), 5–66, 287 (Russian); English translation in Russian Math. Surveys, 32 (1977), 1–65.

  72. — An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. II. Trudy Sem. Petrovsk., (1979), 5–50 (Russian); English translation in Topics in Modern Mathematics, pp. 1–58, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1985.

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

    MATH  MathSciNet  Google Scholar 

  74. Nishida, T., A note on a theorem of Nirenberg. J. Differential Geom., 12 (1977), 629–633 (1978).

    MathSciNet  Google Scholar 

  75. O’Neil, T. M., Collisionless damping of nonlinear plasma oscillations. Phys. Fluids, 8 (1965), 2255–2262.

    Article  MathSciNet  Google Scholar 

  76. O’Neil, T. M. & Coroniti, F.V., The collisionless nature of high-temperature plasmas. Rev. Modern Phys., 71:2 (1999), S404–S410.

    Article  Google Scholar 

  77. Penrose, O., Electrostatic instability of a non-Maxwellian plasma. Phys. Fluids, 3 (1960), 258–265.

    Article  MATH  Google Scholar 

  78. Pfaffelmoser, K., Global classical solutions of the Vlasov–Poisson system in three dimensions for general initial data. J. Differential Equations, 95 (1992), 281–303.

    Article  MATH  MathSciNet  Google Scholar 

  79. Rein, G., Personal communication, 2008.

  80. Robert, R., Statistical mechanics and hydrodynamical turbulence, in Proceedings of the International Congress of Mathematicians (Zürich, 1994), Vol. 2, pp. 1523–1531. Birkhäuser, Basel, 1995.

  81. Ryutov, D. D., Landau damping: half a century with the great discovery. Plasma Phys. Control. Fusion, 41 (1999), A1–A12.

    Article  Google Scholar 

  82. Sáenz, A. W., Long-time behavior of the electric potential and stability in the linearized Vlasov theory. J. Math. Phys., 6 (1965), 859–875.

    Article  Google Scholar 

  83. Saint Raymond, X., A simple Nash–Moser implicit function theorem. Enseign. Math., 35 (1989), 217–226.

    MATH  MathSciNet  Google Scholar 

  84. Schaeffer, J., Global existence of smooth solutions to the Vlasov–Poisson system in three dimensions. Comm. Partial Differential Equations, 16 (1991), 1313–1335.

    Article  MATH  MathSciNet  Google Scholar 

  85. Soffer, A. &Weinstein, M. I., Multichannel nonlinear scattering for nonintegrable equations. Comm. Math. Phys., 133 (1990), 119–146.

    Article  MATH  MathSciNet  Google Scholar 

  86. Spentzouris, L., Ostiguy, J. & Colestock, P., Direct measurement of diffusion rates in high energy synchrotrons using longitudinal beam echoes. Phys. Rev. Lett., 76:4 (1996), 620–623.

    Article  Google Scholar 

  87. Stahl, B., Kiessling, M. K.-H. & Schindler, K., Phase transitions in gravitating systems and the formation of condensed objects. Planet. Space Sci., 43 (1995), 271–282.

    Article  Google Scholar 

  88. Stix, T. H., The Theory of Plasma Waves. McGraw-Hill, New York, 1962.

    MATH  Google Scholar 

  89. Tremaine, S., Hénon, M. & Lynden-Bell, D., H-functions and mixing in violent relaxation. Mon. Not. R. Astr. Soc., 219 (1986), 285–297.

    MATH  Google Scholar 

  90. Turkington, B., Statistical equilibrium measures and coherent states in two-dimensional turbulence. Comm. Pure Appl. Math., 52 (1999), 781–809.

    Article  MathSciNet  Google Scholar 

  91. Vekstein, G. E., Landau resonance mechanism for plasma and wind-generated water waves. Amer. J. Phys., 66:10 (1998), 886–892.

    Article  Google Scholar 

  92. Villani, C., A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, Vol. I, pp. 71–305. North-Holland, Amsterdam, 2002.

  93. — Hypocoercivity. Mem. Amer. Math. Soc., 202 (2009), iv+141.

  94. Wiechen, H., Ziegler, H. J. & Schindler, K., Relaxation of collisionless self-gravitating matter – the lowest energy state. Mon. Not. R. Astr. Soc., 232 (1988), 623–646.

    MATH  Google Scholar 

  95. Zhou, T., Guo, Y. & Shu, C.-W., Numerical study on Landau damping. Phys. D, 157 (2001), 322–333.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WA, UK

    Clément Mouhot

  2. Institut Henri Poincaré & Université de Lyon, Institut Camille Jordan, Université Claude Bernard, 43 Boulevard du 11 novembre 1918, FR-69622, Villeurbanne Cedex, France

    Cédric Villani

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  1. Clément Mouhot
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  2. Cédric Villani
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Correspondence to Cédric Villani.

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Dedicated to Vladimir Arnold and Carlo Cercignani.

AMS Subject Classification: 82C99 (85A05, 82D10).

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Mouhot, C., Villani, C. On Landau damping. Acta Math 207, 29–201 (2011). https://doi.org/10.1007/s11511-011-0068-9

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