zbMATH — the first resource for mathematics

A rigorous stability result for the Vlasov-Poisson system in three dimensions. (English) Zbl 0791.49030
Summary: 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 in \({\mathbf R}^ 3\) with periodic boundary conditions, the space homogeneous stationary solutions \(g\) with energy gradient \(\partial g/\partial\varepsilon\leq 0\) and compact support are (nonlinearly) stable in the \(L^ 1\)-norm with respect to weak solutions of the initial value problem.

49K40 Sensitivity, stability, well-posedness
49J20 Existence theories for optimal control problems involving partial differential equations
Full Text: DOI
[1] Antonov, V. A., Solution of the problem of stability of a stellar system with the Emden density law and spherical velocity distribution, J. Leningrad Univ., no. 7, 135-146 (1962)
[2] Arsen’Ev, A. A., Global existence of a weak solution of Vlasov’s system of equations, U.S.S.R. Comput. Math. Math. Phys., 15, no. 1, 131-143 (1975)
[3] Bardos, C.; Degond, P., Global existence for the Vlasov-Poisson equation in 3space variables with small initial data, Ann. Inst. Henri Poincaré, Analyse non linéaire, 2, 101-118 (1985) · Zbl 0593.35076
[4] Barnes, J.; Goodman, J.; Hut, P., Dynamical instabilities in spherical stellar systems, The Astrophys. J., 300, 112-131 (1986)
[5] Batt, J., Global symmetric solutions of the initial value problem of stellar dynamics, J. Diff. Equations, 25, 342-364 (1977) · Zbl 0366.35020
[6] Batt, J., Asymptotic properties of spherically symmetric self-gravitating mass systems for t → ∞, Transp. Th. Stat. Phys., 16, 763-778 (1987) · Zbl 0645.35010
[7] Batt, J.; Berestycki, H.; Degond, P.; Perthame, B., Some families of solutions of the Vlasov-Poisson system, Arch. Rat. Mech. Anal., 104, 79-103 (1988) · Zbl 0703.35171
[8] Batt, J.; Faltenbacher, W.; Horst, E., Stationary spherically symmetric models in stellar dynamics, Arch. Rat. Mech. Anal., 93, 159-183 (1986) · Zbl 0605.70008
[9] Batt, J.; Pfaffelmoser, K., 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, 499-516 (1988) · Zbl 0676.34017
[10] Baumann, G.; Doremus, J. P.; Feix, M. R., Stability of encounterless spherical stellar systems, Phys. Rev. Lett., 26, 725-728 (1971)
[11] J.Binney - S.Tremaine,Galactic Dynamics, Princeton Series in Astrophysics, Princeton University Press (1987). · Zbl 1130.85301
[12] Di Perna, R. J.; Lions, P.-L., Solutions globales d’équations du type Vlasov-Poisson, C. R. Acad. Sci. Paris, 307, 655-658 (1988) · Zbl 0682.35022
[13] Di Perna, R. J.; Lions, P.-L., Global weak solutions of Vlasov-Maxwell systems, Commun. Pure Appl. Math., 42, 729-757 (1989) · Zbl 0698.35128
[14] A. M.Fridman - V. L.Polyachenko,Physics of Gravitating Systems I, Springer-Verlag (1984). · Zbl 0543.70010
[15] Ganguly, K.; Victory, H. D. Jr., On the convergence of particle methods for multidimensional Vlasov-Poisson systems, SIAM J. Numer. Anal., 26, 249-288 (1989) · Zbl 0669.76146
[16] Glassey, R.; Schaeffer, J., On symmetric solutions of the relativistic Vlasov-Poisson system, Commun. Math. Phys., 101, 459-473 (1985) · Zbl 0582.35110
[17] Glassey, R.; Schaeffer, J., Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data, Commun. Math. Phys., 119, 353-384 (1988) · Zbl 0673.35070
[18] Glassey, R.; Strauss, W., Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rat. Mech. Anal., 92, 59-90 (1986) · Zbl 0595.35072
[19] Glassey, R.; Strauss, W., High velocity particles in a collisionless plasma, Math. Meth. in the Appl. Sci., 9, 46-52 (1987) · Zbl 0649.35079
[20] Glassey, R.; Strauss, W., Absence of shocks in an initially dilute collisionless plasma, Commun. Math. Phys., 113, 191-208 (1987) · Zbl 0646.35072
[21] Hénon, M., Numerical experiments on the stability of spherical stellar systems, Astron. and Astrophys., 24, 229-238 (1973)
[22] Holm, D. D.; Marsden, J. E.; Ratiu, T.; Weinstein, A., Nonlinear stability of fluid and plasma equilibria, Physics Reports, 123, Nos. 1, 1-116 (1985) · Zbl 0717.76051
[23] A.Hörmann,Stabilität beim Vlasov-Poisson-System mit periodischen Feldern, Diplomarbeit, Universität München (1989).
[24] E.Horst,Zum statistischen Anfangswertproblem der Stellardynamik, Diplomarbeit, Universität München (1975).
[25] Horst, E., On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation, Math. Meth. in the Appl. Sci., 3, 229-248 (1981) · Zbl 0463.35071
[26] E.Horst,Global solutions of the relativistic Vlasov-Maxwell system of plasma physics, Habilitationsschrift, Universität München (1986).
[27] Horst, E.; Hunze, R., Weak solutions of the initial value problem for the unmodified nonlinear Vlasov equation, Math. Meth. Appl. in the Sci., 6, 262-279 (1984) · Zbl 0556.35022
[28] Illner, R.; Neunzert, H., An existence theorem for the unmodified Vlasov equation, Math. Meth. Appl. Sci., 1, 530-554 (1979) · Zbl 0415.35076
[29] Kurth, R., A global particular solution to the initial value problem of stellar dynamics, Quarterly Appl. Math., 36, 325-329 (1978) · Zbl 0391.70015
[30] Marchioro, C.; Pulvirenti, M., Some considerations on the non-linear stability of stationary Euler flows, Commun. Math. Phys., 100, 343-354 (1985) · Zbl 0625.76060
[31] Marchioro, C.; Pulvirenti, M., A note on the nonlinear stability of a spatially symmetric Vlasov-Poisson flow, Math. Meth. in the Appl. Sci., 8, 284-288 (1986) · Zbl 0609.35008
[32] K.Pfaffelmoser,Globale klassische Lösungen des dreidimensionalen Vlasov-PoissonSystems, Dissertation, Universität München (1989). · Zbl 0722.35090
[33] M.Reed - B.Simon,Methods of Modern Mathematical Physics, Vol. II, Academic Press (1975). · Zbl 0308.47002
[34] Rein, G., Generic global solutions of the relativistic Vlasov-Maxwell system of plasma physics, Commun. Math. Phys., 135, 41-78 (1990) · Zbl 0722.35091
[35] Schaeffer, J., Global existence for the Poisson-Vlasov system with nearly symmetric data, J. Diff. Equations, 69, 111-148 (1987) · Zbl 0642.35058
[36] Sobouti, Y., Linear oscillations of isotropic stellar systems, Astron. and Astrophys., 140, 82-90 (1984)
[37] Titchmarsh, E. C., Eigenfunction Expansions Associated with Second-Order Differential Equations (1958), Oxford: Clarendon Press, Oxford · Zbl 0097.27601
[38] Whitaker, E. T.; Watson, G. N., A Course in Modern Analysis (1920), Cambridge: University Press, Cambridge
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.