Skip to main content
SpringerLink
Log in

Small BGK Waves and Nonlinear Landau Damping

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Consider a 1D Vlasov-poisson system with a fixed ion background and periodic condition on the space variable. First, we show that for general homogeneous equilibria, within any small neighborhood in the Sobolev space \({W^{s,p}\left( p >1 ,s <1 +\frac{1}{p}\right)}\) of the steady distribution function, there exist nontrivial travelling wave solutions (BGK waves) with arbitrary minimal period and traveling speed. This implies that nonlinear Landau damping is not true in \({W^{s,p}\left( s <1 +\frac{1}{p}\right)}\) space for any homogeneous equilibria and any spatial period. Indeed, in a \({W^{s,p}\left(s <1 +\frac{1}{p}\right)}\) neighborhood of any homogeneous state, the long time dynamics is very rich, including travelling BGK waves, unstable homogeneous states and their possible invariant manifolds. Second, it is shown that for homogeneous equilibria satisfying Penrose’s linear stability condition, there exist no nontrivial travelling BGK waves and unstable homogeneous states in some \({W^{s,p}\left( p >1 ,s >1 +\frac{1}{p}\right)}\) neighborhood. Furthermore, when p = 2, we prove that there exist no nontrivial invariant structures in the \({H^{s}\left( s > \frac{3}{2}\right) }\) neighborhood of stable homogeneous states. These results suggest the long time dynamics in the \({W^{s,p}\left( s >1 +\frac{1}{p}\right) }\) and particularly, in the \({H^{s}\left( s > \frac{3}{2}\right) }\) neighborhoods of a stable homogeneous state might be relatively simple. We also demonstrate that linear damping holds for initial perturbations in very rough spaces, for a linearly stable homogeneous state. This suggests that the contrasting dynamics in W s, p spaces with the critical power \({s=1+\frac{1}{p}}\) is a truly nonlinear phenomena which can not be traced back to the linear level.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Adams R.A., Fournier J.J.F.: Sobolev spaces. Second edition. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam (2003)

    Google Scholar 

  2. Akhiezer, A., Akhiezer, I., Polovin, R., Sitenko, A., Stepanov, K.: Plasma electrodynamics. Vol. I: Linear theory, London: Pergamon Press, 1975 (English Edition, translated by D. ter Haar)

  3. Armstrong T., Montgomery D.: Asymptotic state of the two-stream instability. J. Plasma. Phys. 1(part 4), 425–433 (1967)

    Article  ADS  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Gizzo A., Izrar B., Bertrand P., Fijalkow E., Feix M.R., Shoucri M.: Stability of Bernstein-Greene-Kruskal plasma equilibria. Numerical experiments over a long time. Phys, Fluids 31(1), 72–82 (1988)

    Google Scholar 

  6. Bernstein I., Greene J., Kruskal M.: Exact nonlinear plasma oscillations. Phys. Rev. 108(3), 546–550 (1957)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Bernstein I.B.: Waves in a Plasma in a Magnetic Field. Phys. Rev. 109, 10–21 (1958)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Bohm D., Gross E.P.: Theory of Plasma Oscillations. A. Origin of Medium-Like Behavior. Phys. Rev. 75, 1851–1864 (1949)

    MATH  Google Scholar 

  9. Brunetti M., Califano F., Pegoraro F.: Asymptotic evolution of nonlinear Landau damping. Phys. Rev. E 62, 4109–4114 (2000)

    Article  ADS  Google Scholar 

  10. Buchanan M.L., Dorning J.J.: Nonlinear electrostatic waves in collisionless plasmas. Phys. Rev. E 52, 3015–3033 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  11. Buchanan M.L., Dorning J.J.: Superposition of nonlinear plasma waves. Phys. Rev. Lett. 70, 3732–3735 (1993)

    Article  ADS  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Danielson J.R., Anderegg F., Driscoll C.F.: Measurement of Landau Damping and the Evolution to a BGK Equilibrium. Phys. Rev. Lett. 92, 245003-1–245003-4 (2004)

    Article  ADS  Google Scholar 

  15. Degond P.: Spectral theory of the linearized Vlasov–Poisson equation. Trans. Amer. Math. Soc. 294(2), 435–453 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  16. Demeio L., Zweifel P.F.: Numerical simulations of perturbed Vlasov equilibria. Phys. Fluids B 2, 1252–1255 (1990)

    Article  ADS  Google Scholar 

  17. Demeio L., Holloway J.P.: Numerical simulations of BGK modes. J. Plasma Phys. 46, 63–84 (1991)

    Article  ADS  Google Scholar 

  18. Glassey R., Schaeffer J.: On time decay rates in Landau damping. Comm. Part. Diff. Eqs. 20, 647–676 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  19. Glassey R., Schaeffer J.: Time decay for solutions to the linearized Vlasov equation. Transport Theo. Stat. Phys. 23, 411–453 (1994)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. Guo Y., Strauss W.: Instability of periodic BGK equilibria. Comm. Pure Appl. Math. XLVIII, 861–894 (1995)

    Article  MathSciNet  Google Scholar 

  21. Klimas A.J., Cooper J.: Vlasov–Maxwell and Vlasov–Poisson equations as models of a one-dimensional electron plasma. Phys. Fluids 26, 478–480 (1983)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. Holloway J.P., Dorning J.J.: Undamped plasma waves. Phys. Rev. A 44, 3856–3868 (1991)

    Article  ADS  Google Scholar 

  23. Holloway, J.P., Dorning, J.J.: Nonlinear but small amplitude longitudinal plasma waves. In: Modern mathematical methods in transport theory (Blacksburg, VA, 1989). Oper. Theory Adv. Appl. 51, Basel: Birkhäuser, 1991, pp. 155–179

  24. Hörmander, L.: The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Second edition. Grundlehren der Mathematischen Wissenschaften, 256. Berlin: Springer-Verlag, 1990

  25. Hwang H.J., Vélazquez J.O.: On the existence of exponentially decreasing solutions of the nonlinear landau damping problem. Indiana Univ. Math. J. 58(6), 2623–2660 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Isichenko M.B.: Nonlinear Landau Damping in Collisionless Plasma and Inviscid Fluid. Phys. Rev. Lett. 78, 2369–2372 (1997)

    Article  ADS  Google Scholar 

  27. Krasovsky V.L., Matsumoto H., Omura Y.: Electrostatic solitary waves as collective charges in a magnetospheric plasma: Physical structure and properties of Bernstein–Greene–Kruskal (BGK) solitons. J. Geophys. Res. 108(A3), 1117 (2004)

    Article  Google Scholar 

  28. Lancellotti C., Dorning J.J.: Time-asymptotic wave propagation in collisionless plasmas. Phys. Rev. E 68, 026406 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  29. Landau L.: On the vibration of the electronic plasma. J. Phys. USSR 10, 25 (1946)

    Google Scholar 

  30. Lin Z.: Instability of some ideal plane flows. SIAM J. Math. Anal. 35, 318–356 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  31. Lin Z.: Instability of periodic BGK waves. Math. Res. Letts. 8, 521–534 (2001)

    MATH  Google Scholar 

  32. Lin Z.: Nonlinear instability of periodic waves for Vlasov-Poisson system. Comm. Pure. Appl. Math. 58, 505–528 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  33. Lin, Z., Zeng, C.: Invariant manifolds of Euler equations. Preprint in preparation

  34. Lin, Z., Zeng, C.: Inviscid dynamical structures near Couette flow. Arch. Rat. Mech. Anal., to appear, doi:10.1007/500205-010-0389-9, 2010

  35. Lin, Z., Zeng, C.: Invariant manifolds of Vlasov-Poisson equations. Work in progress

  36. Medvedev M.V., Diamond P.H., Rosenbluth M.N., Shevchenko V.I.: Asymptotic Theory of Nonlinear Landau Damping and Particle Trapping in Waves of Finite Amplitude. Phys. Rev. Lett. 81, 5824 (1998)

    Article  ADS  Google Scholar 

  37. Manfredi G.: Long-Time Behavior of Nonlinear Landau Damping. Phys. Rev. Lett. 79, 2815 (1997)

    Article  ADS  Google Scholar 

  38. Mouhot, C., Villani, C.: On Landau damping. Acta Math. (to appear)

  39. Muschietti L., Ergun R.E., Roth I., Carlson C.W.: Phase-space electron holes along magnetic field lines. Geophys. Res. Lett. 26, 1093–1096 (1999)

    Article  ADS  Google Scholar 

  40. Orr W.McF.: Stability and instability of steady motions of a perfect liquid. Proc. Ir. Acad. Sect. A, Math Astron. Phys. Sci. 27, 9–66 (1907)

    Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  43. Stein E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, NJ (1970)

    Google Scholar 

  44. Strichartz R.S.: Multipliers on fractional Sobolev spaces. J. Math. Mech. 16, 1031–1060 (1967)

    MATH  MathSciNet  Google Scholar 

  45. Tartar L.: An introduction to Sobolev spaces and interpolation spaces. Lecture Notes of the Unione Matematica Italiana, 3. Springer/UMI, Berlin-Bologna (2007)

    Google Scholar 

  46. Triebel H.: Theory of function spaces. Monographs in Mathematics, 78. Birkhäuser Verlag, Basel (1983)

    Google Scholar 

  47. Valentini F., Carbone V., Veltri P., Mangeney A.: Wave-Particle Interaction and Nonlinear Landau Damping in Collisionless Electron Plasmas. Transport Th. Stat. Phys. 34, 89–101 (2005)

    Article  ADS  MATH  Google Scholar 

  48. Weitzner H.: Plasma oscillations and Landau damping. Phys. Fluids 6, 1123–1127 (1963)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  50. Zhou T., Guo Y., Shu C.-W.: Numerical study on Landau damping. Physica D 157, 322–333 (2001)

    Article  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Zhiwu Lin & Chongchun Zeng

Authors
  1. Zhiwu Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chongchun Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhiwu Lin.

Additional information

Communicated by H. Spohn

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lin, Z., Zeng, C. Small BGK Waves and Nonlinear Landau Damping. Commun. Math. Phys. 306, 291–331 (2011). https://doi.org/10.1007/s00220-011-1246-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-011-1246-5

Keywords

Search

Navigation