On Global Solutions And Asymptotic Behavior Of Two Classes Of Vlasov-Poisson Systems

Vlasov-Poisson system, which comes from statistical mechanics, plays an important role in the field of mathematical physics equation. It is used to describe how particle system move under self-consistent Newton force or Coulomb force, based on which, macroscopic physical properties of particle system is obtained. In this paper, we focus on the well-posedness and asymptotic behavior of the Vlasov-Poisson system with radiation damping and Vlasov-Poisson system with point charges in three-dimensional space.In the first chapter, we mainly introduce the background, method and status of re-searches on Vlasov-Poisson system, Vlasov-Poisson system with radiation damping and Vlasov-Poisson system with point charges. The contribution of this paper is also presented in this chapter.Chapter2and chapter3are devoted to study the Vlasov-Poisson system with radiation damping. Due to the strong nonlinearity of damping D[3](t), it is hard to establish some valuable estimation of kinetic formulas (such as the boundedness of kinetic energy), which makes it difficult to prove the existence of global classical solution of Cauchy problem of Vlasov-Poisson system with radiation damping. In chapter2, under the condition of finite mass, finite inertia and finite kinetic energy of initial value, we demonstrate the global existence of nonnegative weak solutions and asymptotic behavior of the Cauchy problem of the Vlasov-Poisson system with correction damping[46]. In chapter3, with dispersion property of the macroscopic density and Schauder’s fixed point theorem, we establish the existence of a global classical solution to the damped Vlasov-Poisson system under the assumption that initial value is sufficiently small.Chapter4and chapter5focus on the Cauchy problem of the Vlasov-Poisson system with point charges. When the point charges have same (opposite) sign with the plasma, there is a repulsive interaction (attractive interaction) between them. In chapter4, for the attractive interaction case, we define a weak solution based on Diperna-Lions flow and establish bounds of kinetic energy by introducing a Lyapunov functional. Based on these results, we establish global existence of weak solutions for this system. In the repulsive case, we obtain a polynomial growth estimate of the size of velocity support for compactly supported classical solutions in chapter5, which improves an existing results on exponential growth estimateIn the end, we summarize our works and discuss some interesting open problems for Vlasov-Poisson system, Vlasov-Poisson system with radiation damping and Vlasov-Poisson system with point charges.