Acta Mathematica Sinica, English Series

, Volume 33, Issue 5, pp 635–656 | Cite as

Existence, uniqueness and asymptotic behavior for the Vlasov–Poisson system with radiation damping

  • Jing Chen
  • Xian Wen Zhang
  • Ran Gao


We investigate the Cauchy problem for the Vlasov–Poisson system with radiation damping. By virtue of energy estimate and a refined velocity average lemma, we establish the global existence of nonnegative weak solution and asymptotic behavior under the condition that initial data have finite mass and energy. Furthermore, by building a Gronwall inequality about the distance between the Lagrangian flows associated to the weak solutions, we can prove the uniqueness of weak solution when the initial data have a higher order velocity moment.


Vlasov–Poisson system radiation damping velocity averages weak solution uniqueness 

MR(2010) Subject Classification

35Q83 35L60 82C21 82D10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors would like to thank anonymous referees for their valuable suggestions concerning the presentation of this paper.


  1. [1]
    Aubin, J. P.: Un théorème de compacité. C. R. Acad. Sci. Paris Sér. I Math., 256, 5042–5044 (1963)zbMATHGoogle Scholar
  2. [2]
    Bauer, S., Kunze, M.: Radiative friction for charges interacting with the radiation field: Classical manyparticle systems, in Analysis, Modeling and Simulation of Multiscale Problems, Springer, Berlin, 2006Google Scholar
  3. [3]
    Bouchut, F., Desvillettes, L.: Averaging lemmas without time Fourier transform and application to discretized kinetic equations. Proc. Roy. Soc. Edinburgh Sect. A, 148, 19–36 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Diperna, R. J., Lions, P. L.: Global weak solutions of Vlasov–Maxwell system. Comm. Pure Appl. Math., 42, 729–757 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Evelyne, M.: A uniqueness criterion for unbounded solutions to the Vlasov–Poisson system. Commun. Math. Phys., 346, 469–482 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Horst, E., Hunze, R.: Weak solutions of the inintial valume problem for the unmodified nonlinear Vlasov equation. Math. Methods Appl. Sci., 6, 262–279 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Kunze, M., Rendall, A. D.: The Vlasov–Poisson system with radiation damping. Ann. Henri Poincaré, 2, 857–886 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Lions, P. L., Perthame, B.: Propagation of moments and regularity for the 3-dimensional Vlasov–Poisson system. Invent. Math., 105, 415–430 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Lieb, E. H., Loss, M.: “Analysis”, American Mathematical Society, Providence, RI, 2001zbMATHGoogle Scholar
  10. [10]
    Loeper, G.: Uniqueness of the solution to the Vlasov–Poisson system with bounded density. J. Math. Pures Appl., 86, 68–79 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Okabe, S., Ukai, T.: On classical solutions in the large in time of the two-dimensional Vlasov equation. Osaka, J. Math., 15, 245–261 (1978)MathSciNetzbMATHGoogle Scholar
  12. [12]
    Perthame, B.: Time decay, propagation of low moments and dispersive effects for kinetic equations. Comm. Partial Differential Equations, 21, 659–686 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Pfaffelmoser, K.: Global existence of the Vlasov–Poisson system in three dimensions for general initial data. J. Differ. Equ., 95, 281–303 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Rein. G.: Collisionless kinetic equation from astrophysics — The Vlasov–Poisson system, in Handbook of Differential Equations: Evolutionary Equations, Vol. 3, eds. C.M. Dafermos and E. Feireisl, Elsevier, Amsterdam, 2007, 383–476CrossRefGoogle Scholar
  15. [15]
    Stein, E. M.: Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970zbMATHGoogle Scholar
  16. [16]
    Zhang, X., Wei, J.: The Vlasov–Poisson system with infinite kinetic energy and initial data in L p(R6). J. Math. Anal. Appl., 341, 548–558 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Zhang, X.: Global weak solutions to the cometary flow equation with a self-generated electric field. J. Math. Anal. Appl., 377, 593–612 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.College of ScienceZhongyuan University of TechnologyZhengzhouP. R. China
  2. 2.School of Mathematics and StatisticsHuazhong University of Science and TechnologyWuhanP. R. China

Personalised recommendations