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A note on the nonlinear stability of a spatially symmetric Vlasov-Poisson flow. (English) Zbl 0609.35008
The Vlasov-Poisson equation was used to model a continuum of electrons moving in a \(\nu\)-dimensional flat torus. The Coulomb interaction was assumed for charged particles. The system is neutralized by a spatially uniform, positively charged background. The authors studied the problem of nonlinear stability of a special class of stationary solutions with respect to perturbations of the initial datum. Sufficient conditions were given for the homogeneous case with non-smooth solutions. This indicates that the constants appearing in the stability result do not depend on the regularity of the problem.
Reviewer: L.Shih

35B35 Stability in context of PDEs
82D10 Statistical mechanical studies of plasmas
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
35Q99 Partial differential equations of mathematical physics and other areas of application
Full Text: DOI
[1] Arsenev, Global existence of a weak solution of Vlasov’s system. URRS Computational, Math. and Mech. Phys. 15 pp 131– (1975)
[2] Bardos, Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data. Ann. Inst. H. Poincaré:, Analyse non linéaire 2 (2) pp 101– (1985) · Zbl 0593.35076
[3] Batt, Global symmetric solutions of the initial value problem of stellar dynamics, J. Diff. Eq. 25 pp 342– (1977) · Zbl 0366.35020 · doi:10.1016/0022-0396(77)90049-3
[4] Holm, Nonlinear stability of fluid and plasma equilibria, Physics Reports 123 pp 1– (1985) · Zbl 0717.76051 · doi:10.1016/0370-1573(85)90028-6
[5] Horst, On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation, Math. Meth. in the Appl. Sci. 3 pp 229– (1981) · Zbl 0463.35071 · doi:10.1002/mma.1670030117
[6] Math. Meth. in the Appl. Sci. 4 pp 19– (1982) · Zbl 0485.35079 · doi:10.1002/mma.1670040104
[7] Illner, An existence theorem for the unmodified vlasov equation, Math. Meth. in the Appl. Sci. 1 pp 530– (1979) · Zbl 0415.35076 · doi:10.1002/mma.1670010410
[8] Iordanskii, The Cauchy problem for the kinetic equation of plasmas, Amer. Math. Soc. Trans. Ser. 35 pp 351– (1964) · Zbl 0127.21902 · doi:10.1090/trans2/035/12
[9] Marchioro, Some considerations on the nonlinear stability of stationary planar flows, Comm. Math. Phys. 100 pp 343– (1985) · Zbl 0625.76060 · doi:10.1007/BF01206135
[10] Neunzert, Lectures Notes in Math 10948 pp 60– (1984)
[11] Ukai, On the classical solutions in large in time of the two-dimensional Vlasov equation. Osaka, J. Math. 15 pp 245– (1978) · Zbl 0405.35002
[12] Wollman , S. Existence and uniqueness theory of the Vlasov equation. 1982 · Zbl 0506.45012
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