Abstract
It is proven that in a neutral two-component plasma with space homogeneous positively charged background, which is governed by the Vlasov-Poisson system and for which Poisson's equation is considered on a cube inR 3 with periodic boundary conditions, the space homogeneous stationary solutions g with energy gradient ∂g/∂ε ≤ 0 and compact support are (nonlinearly) stable in the L1-norm with respect to weak solutions of the initial value problem.
Article PDF
Similar content being viewed by others
References
V. A. Antonov,Solution of the problem of stability of a stellar system with the Emden density law and spherical velocity distribution, J. Leningrad Univ., no.7, Ser. Math., Mekh. Astro., no.2 (1962), pp. 135–146.
A. A. Arsen'ev,Global existence of a weak solution of Vlasov's system of equations, U.S.S.R. Comput. Math. Math. Phys.,15, no. 1 (1975), pp. 131–143.
C. Bardos -P. Degond,Global existence for the Vlasov-Poisson equation in 3space variables with small initial data, Ann. Inst. Henri Poincaré, Analyse non linéaire,2 (1985), pp. 101–118.
J. Barnes -J. Goodman -P. Hut,Dynamical instabilities in spherical stellar systems, The Astrophys. J.,300 (1986), pp. 112–131.
J. Batt,Global symmetric solutions of the initial value problem of stellar dynamics, J. Diff. Equations,25 (1977), pp. 342–364.
J. Batt,Asymptotic properties of spherically symmetric self-gravitating mass systems for t → ∞, Transp. Th. Stat. Phys.,16 (1987), pp. 763–778.
J. Batt -H. Berestycki -P. Degond -B. Perthame,Some families of solutions of the Vlasov-Poisson system, Arch. Rat. Mech. Anal.,104 (1988), pp. 79–103.
J. Batt -W. Faltenbacher -E. Horst,Stationary spherically symmetric models in stellar dynamics, Arch. Rat. Mech. Anal.,93 (1986), pp. 159–183.
J. Batt -K. Pfaffelmoser,On the radius continuity of the models of polytropic gas spheres which correspond to the positive solutions of the generalized Emden-Fowler equation, Math. Meth. Appl. in the Sci.,10 (1988), pp. 499–516.
G. Baumann -J. P. Doremus -M. R. Feix,Stability of encounterless spherical stellar systems, Phys. Rev. Lett.,26 (1971), pp. 725–728.
J.Binney - S.Tremaine,Galactic Dynamics, Princeton Series in Astrophysics, Princeton University Press (1987).
R. J. Di Perna -P.-L. Lions,Solutions globales d'équations du type Vlasov-Poisson, C. R. Acad. Sci. Paris,307, Série I (1988), pp. 655–658.
R. J. Di Perna -P.-L. Lions,Global weak solutions of Vlasov-Maxwell systems, Commun. Pure Appl. Math.,42 (1989), pp. 729–757.
A. M.Fridman - V. L.Polyachenko,Physics of Gravitating Systems I, Springer-Verlag (1984).
K. Ganguly -H. D. Victory Jr.,On the convergence of particle methods for multidimensional Vlasov-Poisson systems, SIAM J. Numer. Anal.,26 (1989), pp. 249–288.
R. Glassey -J. Schaeffer,On symmetric solutions of the relativistic Vlasov-Poisson system, Commun. Math. Phys.,101 (1985), pp. 459–473.
R. Glassey -J. Schaeffer,Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data, Commun. Math. Phys.,119 (1988), pp. 353–384.
R. Glassey -W. Strauss,Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rat. Mech. Anal.,92 (1986), pp. 59–90.
R. Glassey -W. Strauss,High velocity particles in a collisionless plasma, Math. Meth. in the Appl. Sci.,9 (1987), pp. 46–52.
R. Glassey -W. Strauss,Absence of shocks in an initially dilute collisionless plasma, Commun. Math. Phys.,113 (1987), pp. 191–208.
M. Hénon,Numerical experiments on the stability of spherical stellar systems, Astron. and Astrophys.,24 (1973), pp. 229–238.
D. D. Holm -J. E. Marsden -T. Ratiu -A. Weinstein,Nonlinear stability of fluid and plasma equilibria, Physics Reports,123, Nos. 1 and 2 (1985), pp. 1–116.
A.Hörmann,Stabilität beim Vlasov-Poisson-System mit periodischen Feldern, Diplomarbeit, Universität München (1989).
E.Horst,Zum statistischen Anfangswertproblem der Stellardynamik, Diplomarbeit, Universität München (1975).
E. Horst,On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation, Part I and II, Math. Meth. in the Appl. Sci.,3 (1981), pp. 229–248;4 (1982), pp. 19–32.
E.Horst,Global solutions of the relativistic Vlasov-Maxwell system of plasma physics, Habilitationsschrift, Universität München (1986).
E. Horst -R. Hunze,Weak solutions of the initial value problem for the unmodified nonlinear Vlasov equation, Math. Meth. Appl. in the Sci.,6 (1984), pp. 262–279.
R. Illner -H. Neunzert,An existence theorem for the unmodified Vlasov equation, Math. Meth. Appl. Sci.,1 (1979), pp. 530–554.
R. Kurth,A global particular solution to the initial value problem of stellar dynamics, Quarterly Appl. Math.,36 (1978), pp. 325–329.
C. Marchioro -M. Pulvirenti,Some considerations on the non-linear stability of stationary Euler flows, Commun. Math. Phys.,100 (1985), pp. 343–354.
C. Marchioro -M. Pulvirenti,A note on the nonlinear stability of a spatially symmetric Vlasov-Poisson flow, Math. Meth. in the Appl. Sci.,8 (1986), pp. 284–288.
K.Pfaffelmoser,Globale klassische Lösungen des dreidimensionalen Vlasov-PoissonSystems, Dissertation, Universität München (1989).
M.Reed - B.Simon,Methods of Modern Mathematical Physics, Vol. II, Academic Press (1975).
G. Rein,Generic global solutions of the relativistic Vlasov-Maxwell system of plasma physics, Commun. Math. Phys.,135 (1990), pp. 41–78.
J. Schaeffer,Global existence for the Poisson-Vlasov system with nearly symmetric data, J. Diff. Equations,69 (1987), pp. 111–148.
Y. Sobouti,Linear oscillations of isotropic stellar systems, Astron. and Astrophys.,140 (1984), pp. 82–90.
E. C. Titchmarsh,Eigenfunction Expansions Associated with Second-Order Differential Equations, Part II, Clarendon Press, Oxford (1958).
E. T. Whitaker -G. N. Watson,A Course in Modern Analysis, University Press, Cambridge (1920).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Batt, J., Rein, G. A rigorous stability result for the Vlasov-Poisson system in three dimensions. Annali di Matematica pura ed applicata 164, 133–154 (1993). https://doi.org/10.1007/BF01759319
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01759319