On the uniqueness and stability of weak solutions of a fokker-planck-vlasov equation

  • Reimund Rautmann
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 679)


In [7] the existence of weak solutions of a Fokker-Planck-Vlasov equation is proved. In this paper, with a little more stringent assumption we show the uniqueness of weak solutions and establish a criterion of (asymptotic) stability against local disturbations.-As a consequence, in the case of uniqueness the Galerkin method used in [7] is a constructive one, i.e. the whole sequence of all Galerkin-approximations converges.


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  1. 1.
    HANNOSCHÖCK, G., Existenz und Eindeutigkeit bei der Fokker-Planck-Gleichung mit modifiziertem Vlasov-Term, unpublished.Google Scholar
  2. 2.
    HOPF, E., Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951), 213–231.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    LADYŽENSKAJA, O.A., SOLONNIKOV, V.A., URAL’CEVA, N.N.: Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, Rhode Island (1968).Google Scholar
  4. 4.
    RAUTMANN, R., Bemerkungen zur Anfangswertaufgabe einer stabilisierten Navier-Stokesschen Gleichung, ZAMM 55 (1975), T 217–221.MathSciNetGoogle Scholar
  5. 5.
    RAUTMANN, R., On the Convergence of a Galerkin Method to Solve the Initial Value Problem of a Stabilized Navier-Stokes Equation, ISNM 27 Birkhäuser Verlag, Basel, Stuttgart (1975), 255–264.MATHGoogle Scholar
  6. 6.
    RAUTMANN, R., Ein kenvergentes Hopf-Galerkin-Verfahren für eine Gleichung vom FOKKER-PLANCK-Typ, ZAMM 57 (1977), T 252–253.MathSciNetGoogle Scholar
  7. 7.
    RAUTMANN, R., The Existence of Weak Solutions of the Fokker-Planck-Vlasov-Equation, to appear in: Methoden und Verfahren d. Math. Physik.Google Scholar
  8. 8.
    SERRIN, J., The Initial Value Problem for the Navier-Stokes Equations, in: Nonlinear Problems (ed. R.E. Langer) MRC Madison (1963), 69–98.Google Scholar
  9. 9.
    WALTER, W.: Differential and Integral Inequalities, Springer Berlin (1970).Google Scholar
  10. 10.
    WAX, N. (ed.) Selected Papers on Noise and Stochastic Processes, Dover, Publ., Inc. New York (1954).MATHGoogle Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • Reimund Rautmann

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