Archive for Rational Mechanics and Analysis

, Volume 183, Issue 3, pp 489–524

N-particles Approximation of the Vlasov Equations with Singular Potential



We prove the convergence in any time interval of a point-particle approximation of the Vlasov equation by particles initially equally separated for a force in 1/|x|α, with \(\alpha \leqq 1\). We introduce discrete versions of the L norm and time averages of the force-field. The core of the proof is to show that these quantities are bounded and that consequently the minimal distance between particles in the phase space is bounded from below.


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  1. 1.
    Batt, J.: N-Particle approximation to the nonlinear Vlasov–Poisson system. PreprintGoogle Scholar
  2. 2.
    Bouchut F. (1995). Smoothing effect for the non-linear Vlasov–Poisson–Fokker–Planck system. J. Differential Equations 122:225–238CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Batt J., Rein G. (1991). Global classical solutions of the periodic Vlasov–Poisson system in three dimensions. C.R. Math. Acad. Sci. Paris 313:411–416MathSciNetMATHGoogle Scholar
  4. 4.
    Braun W., Hepp K. (1977). The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles. Comm. Math. Phys. 56:101–113CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Cercignani C., Illner R., Pulvirenti M. (1994). The Mathematical Theory of Dilute Gases. Applied Mathematical Sciences, 106, Springer-Verlag, New YorkMATHGoogle Scholar
  6. 6.
    Dobrušin R.L. (1979). Vlasov equations. Funktsional. Anal. i Prilozhen. 13:48–58MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gasser I., Jabin P.E., Perthame B. (2000). Regularity and propagation of moments in some nonlinear Vlasov systems. Proc. Roy. Soc. Edinburgh Sect. A 130:1259–1273MathSciNetMATHGoogle Scholar
  8. 8.
    Glassey R.T. (1996). The Cauchy Problem in Kinetic Theory. SIAM, Philadelphia PAMATHGoogle Scholar
  9. 9.
    Goodman J., Hou T.Y., Lowengrub J. (1990). Convergence of the point vortex method for the 2-D Euler equations. Comm. Pure Appl. Math. 43:415–430MathSciNetMATHGoogle Scholar
  10. 10.
    Horst E. (1981). On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation I. Math. Methods Appl. Sci. 3:229–248MathSciNetMATHADSGoogle Scholar
  11. 11.
    Horst E. (1982). On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation II. Math. Methods Appl. Sci. 4:19–32ADSMathSciNetMATHGoogle Scholar
  12. 12.
    Illner R., Pulvirenti M. (1989). Global validity of the Boltzmann equation for two and three-dimensional rare gas in vacuum. Comm. Math. Phys. 121:143–146CrossRefADSMathSciNetMATHGoogle Scholar
  13. 13.
    Jabin P.E., Otto F. (2004). Identification of the dilute regime in particle sedimentation. Comm. Math. Phys. 250:415–432CrossRefADSMathSciNetMATHGoogle Scholar
  14. 14.
    Jabin, P.E., Perthame, B.: Notes on mathematical problems on the dynamics of particles interacting through a fluid. Modelling in Applied Sciences (Ed. Bellomo, P., Pulvirenti, M.), pp. 111–147, Modelling and Simulation in Science, Engineering and Technology, Birkhauser, Boston, 2000Google Scholar
  15. 15.
    Lions P.L., Perthame B. (1991). Propagation of moments and regularity for the 3-dimensional Vlasov–Poisson System. Invent. Math. 105:415–430CrossRefADSMathSciNetMATHGoogle Scholar
  16. 16.
    Neunzert, H., Wick, J.: Theoretische und numerische Ergebnisse zur nichtlinearen Vlasov–Gleichung. Numerische Lösung nichtlinearer partieller Differential- und Integrodifferentialgleichungen (Tagung, Math. Forschungsinst., Oberwolfach, 1971), pp. 159–185. Lecture Notes in Mathematics, Vol. 267, Springer, Berlin, 1972Google Scholar
  17. 17.
    Perthame B. (1996). Time decay, propagation of low moments and dispersive effects for kinetic equations. Comm. Partial Differential Equations 21:659–686MathSciNetMATHGoogle Scholar
  18. 18.
    Pfaffelmoser K. (1992). Global classical solutions of the Vlasov–Poisson system in three dimensions for general initial data. J. Differential Equations 95:281–303CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Pulvirenti M., Simeoni C. (2000). L -estimates for the Vlasov–Poisson–Fokker–Planck Equation. Math. Methods Appl. Sci. 23:923–935CrossRefMathSciNetMATHADSGoogle Scholar
  20. 20.
    Schaeffer J. (1991). Global existence of smooth solutions to the Vlasov–Poisoon system in three dimensions. Comm. Partial Differential Equations 16:1313–1335MathSciNetMATHGoogle Scholar
  21. 21.
    Schochet S. (1995). The weak vorticity formulation of the 2-D Euler equations and concentration-cancellation. Comm. Partial Differential Equations 20:1077–1104MathSciNetMATHGoogle Scholar
  22. 22.
    Schochet S. (1996). The point-vortex method for periodic weak solutions of the 2-D Euler equations. Comm. Pure Appl. Math. 49:911–965CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Spohn, H.: Large Scale Dynamics of Interacting Particles. Springer-Verlag Berlin 1991Google Scholar
  24. 24.
    Victory H.D. jr., Allen E.J. (1991). The convergence theory of particle-in-cell methods for multidimensional Vlasov–Poisson systems. SIAM J. Numer. Anal. 28:1207–1241CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Wollman S. (2000). On the approximation of the Vlasov–Poisson system by particles methods. SIAM J. Numer. Anal. 37:1369–1398CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Wollman S. (1993). Global in time solutions to the three-dimensional Vlasov–Poisson System. J. Math. Anal. Appl. 176:76–91CrossRefMathSciNetMATHGoogle Scholar

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© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis Lions, U.M.R. no 7598 du C.N.R.S.Université Pierre et Marie CurieParis Cedex 05France
  2. 2.Laboratoire J.-A. Dieudonn, U.M.R. no 6621 du C.N.R.S.Université de Nice – Sophia Antipolis Parc ValroseNice Cedex 02France

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