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The Vlasov-Poisson-Landau System with a Uniform Ionic Background and Algebraic Decay Initial Perturbation
by Yuanjie Lei   Ling Wan   Huijiang Zhao

Vol. 10 No. 3 (2015) P.311~P.347

ABSTRACT

In the absence of magnetic effects, the dynamics of two-species charged dilute particles (e.g., electrons and ions) interacting with their self-consistent electrostatic field as well as their grazing collisions is described by the two-species Vlasov-Poisson-Landau system, while the one-species Vlasov-Poisson-Landau system models the time evolution of dilute charged particles consisting of electrons interacting through its binary grazing collisions under the influence of the self-consistent internally generated electrostatic forces with a fixed ionic background. To construct global smooth solutions of the two-species Vlasov-Poisson- Landau system near Maxwellians, a time-velocity weighted energy method is developed by Guo in [Guo Y., $\it{J. Amer. Math. Soc.}$ $\bf{25}$ (2012), 759~812] which yields a satisfactory well-posedness theory for the two-species Vlasov-Poisson-Landau system with algebraic decay initial perturbation in the perturbative context. It is worth emphasizing that such a time-velocity weighted energy method relies heavily on the fact that the potential of the electrostatic field decays sufficiently fast. The main purpose of this paper is to show that, for the one-species Vlasov-Poisson-Landau system, although the temporal decay of the electric potential is worse than that of the two-species Vlasov-Poisson-Landau system, the method devel-oped in [Guo Y., $\it{J. Amer. Math. Soc.}$ $\bf{25}$ (2012), 759~812] can still be adapted provided that the initial perturbation satisfies the neutral condition.

KEYWORDS
One-species Vlasov-Poisson-Landau system; global solutions near Maxwellians; algebraic decay initial perturbation; neutral condition.

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 35Q83 (35A01; 35A02; 35B40).

MILESTONES