Skip to main content
Log in

Landau Damping for Vlasov–Poisson System with Radiation Damping on the Torus

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

Abstract

This paper studies the nonlinear Landau damping on the torus \(\mathbb {T}^3\) for the Vlasov–Poisson system with radiation damping. We show that even under the influence of the radiation damping effects, the plasma still has the remarkable property of Landau damping. The corresponding density and force field decay exponentially fast as time goes to infinity.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

Data Availability

No datasets were generated or analysed during the current study.

References

  • Batt, J.: Ein Existenzbeweis für die Vlasov-Gleichung der Stellar-dynamik bei gemittelter Dichte. Arch. Ration. Mech. Anal. 13, 296–308 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  • Batt, J.: Global symmetric solutions of the initial value problem of stellar dynamics. J. Differ. Equ. 25(3), 342–364 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  • 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(6), 411–416 (1991)

    MathSciNet  MATH  Google Scholar 

  • Bauer, S., Kunze, M.: Radiative friction for charges interacting with the radiation field: classical many-particle systems. In: Analysis. Modeling and Simulation of Multiscale Problems, pp. 531–551. Springer, Berlin (2006)

  • Bedrossian, J., Masmoudi, N., Mouhot, C.: Landau damping: paraproducts and Gevrey regularity. Ann. PDE, 2(1):Art. 4, 71 (2016)

  • Bedrossian, J., Masmoudi, N., Mouhot, C.: Landau damping in finite regularity for unconfined systems with screened interactions. Commun. Pure Appl. Math. 71(3), 537–576 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  • Bedrossian, J., Masmoudi, N., Mouhot, C.: Linearized wave-damping structure of Vlasov–Poisson in \(\mathbb{R} ^3\). SIAM J. Math. Anal. 54(4), 4379–4406 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  • Castella, F.: Propagation of space moments in the Vlasov–Poisson equation and further results. Ann. Inst. H. Poincaré Anal. Non Linéaire 16(4), 503–533 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, J., Zhang, X.W., Gao, R.: Existence, uniqueness and asymptotic behavior for the Vlasov–Poisson system with radiation damping. Acta Math. Sin. (Engl. Ser.) 33(5), 635–656 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, Z., Zhang, X.: Sub-linear estimate of large velocities in a collisionless plasma. Commun. Math. Sci. 12(2), 279–291 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Duan, R., Zhang, Z.: A note on Landau damping of two-species Vlasov–Poisson system. arXiv preprint arXiv:2407.02822 (2024)

  • Gagnebin, A., Iacobelli, M.: Landau damping on the torus for the Vlasov–Poisson system with massless electrons. J. Differ. Equ. 376, 154–203 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  • Glassey, R., Schaeffer, J.: Time decay for solutions to the linearized Vlasov equation. Transp. Theory Stat. Phys. 23(4), 411–453 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  • Glassey, R.T.: The Cauchy problem in kinetic theory. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1996)

    Book  MATH  Google Scholar 

  • Grenier, E., Nguyen, T.T., Rodnianski, I.: Landau damping for analytic and Gevrey data. Math. Res. Lett. 28(6), 1679–1702 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Han-Kwan, D., Nguyen, T. T., Rousset, F.: Asymptotic stability of equilibria for screened Vlasov–Poisson systems via pointwise dispersive estimates. Ann. PDE, 7(2):Paper No. 18, 37, (2021)

  • Han-Kwan, D., Nguyen, T.T., Rousset, F.: On the linearized Vlasov–Poisson system on the whole space around stable homogeneous equilibria. Commun. Math. Phys. 387(3), 1405–1440 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Horst, E.: On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation. II. Special Cases. Math. Methods Appl. Sci. 4(1), 19–32 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  • Horst, E.: On the asymptotic growth of the solutions of the Vlasov–Poisson system. Math. Methods Appl. Sci. 16(2), 75–86 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Illner, R., Rein, G.: Time decay of the solutions of the Vlasov–Poisson system in the plasma physical case. Math. Methods Appl. Sci. 19(17), 1409–1413 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Ionescu, A. D., Pausader, B., Wang, X., Widmayer, K.: Nonlinear Landau damping for the Vlasov–Poisson system in \(\mathbb{R}^3\): the Poisson equilibrium. Ann. PDE, 10(1):Paper No. 2, 78 (2024)

  • Kunze, M., Rendall, A.D.: The Vlasov–Poisson system with radiation damping. Ann. Henri Poincaré 2(5), 857–886 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Landau, L.: On the vibrations of the electronic plasma. Akad. Nauk SSSR. Zhurnal Eksper. Teoret. Fiz. 16, 574–586 (1946)

    MathSciNet  MATH  Google Scholar 

  • Lin, Z., Zeng, C.: Small BGK waves and nonlinear Landau damping. Commun. Math. Phys. 306(2), 291–331 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Lions, P.-L., Perthame, B.: Propagation of moments and regularity for the \(3\)-dimensional Vlasov–Poisson system. Invent. Math. 105(2), 415–430 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • Loeper, G.: Uniqueness of the solution to the Vlasov–Poisson system with bounded density. J. Math. Pures Appl. (9) 86(1), 68–79 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Mouhot, C., Villani, C.: On Landau damping. Acta Math. 207(1), 29–201 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Pallard, C.: Growth estimates and uniform decay for a collisionless plasma. Kinet. Relat. Models 4(2), 549–567 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Pallard, C.: A note on the growth of velocities in a collisionless plasma. Math. Methods Appl. Sci. 34(7), 803–806 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Pallard, C.: Large velocities in a collisionless plasma. J. Differ. Equ. 252(3), 2864–2876 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Pallard, C.: Moment propagation for weak solutions to the Vlasov–Poisson system. Comm. Partial Differ. Equ. 37(7), 1273–1285 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Pallard, C.: Space moments of the Vlasov–Poisson system: propagation and regularity. SIAM J. Math. Anal. 46(3), 1754–1770 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Perthame, B.: Time decay, propagation of low moments and dispersive effects for kinetic equations. Commun. Partial Differ. Equ. 21(3–4), 659–686 (1996)

    MathSciNet  MATH  Google Scholar 

  • Pfaffelmoser, K.: Global classical solutions of the Vlasov–Poisson system in three dimensions for general initial data. J. Differ. Equ. 95(2), 281–303 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Rein, G.: Growth estimates for the solutions of the Vlasov–Poisson system in the plasma physics case. Math. Nachr. 191, 269–278 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Schaeffer, J.: Asymptotic growth bounds for the Vlasov–Poisson system. Math. Methods Appl. Sci. 34(3), 262–277 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Xiao, M., Zhang, X.: On global solutions to the Vlasov–Poisson system with radiation damping. Kinet. Relat. Models 11(5), 1183–1209 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  • Xiao, M., Zhang, X.: Moment propagation of the Vlasov–Poisson system with a radiation term. Acta Appl. Math. 160, 185–206 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang, X., Wei, J.: The Vlasov–Poisson system with infinite kinetic energy and initial data in \(L^p(\mathbb{R} ^6)\). J. Math. Anal. Appl. 341(1), 548–558 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to warmly thank the anonymous referee for the constructive review and the useful comments that improved the presentation of the results.

Funding

This work is supported by the National Natural Science Foundation of China (Grant No.11871024).

Author information

Authors and Affiliations

Authors

Contributions

Z.W. wrote the main manuscript text and conducted most of calculations in the manuscript. X.Z. participated in most of the theoretical analysis and carefully revised the presentation of the manuscript. All authors reviewed the manuscript.

Corresponding author

Correspondence to Xianwen Zhang.

Ethics declarations

Conflict of interest

No conflict of interest.

Additional information

Communicated by Pierre Degond.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Z., Zhang, X. Landau Damping for Vlasov–Poisson System with Radiation Damping on the Torus. J Nonlinear Sci 35, 30 (2025). https://doi.org/10.1007/s00332-024-10126-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00332-024-10126-3

Keywords

Mathematics Subject Classification

Navigation