Qualitative Research On The Vlasov-Poisson(Klein-Gordon) System

Recently,more and more researchers pay attention to the research of Kinetic equation,due to it involves many significant phenomena and area of science,such as,nuclear reactor,electromagnetic radiation,plasma physics,mathematical theory of dilute gases,astrophysic-s,aviation and so on.This paper is devoted to making some qualitative researches on the Vlasov-Poisson system and relativistic Vlasov-Klein-Gordon system.The first chapter mainly elaborates the background,evolution,relation and status of re-searches on the Vlasov-Poisson system,the relativistic Vlasov-Maxwell system,the Vlasov-Poisson system with radiation damping and the relativistic Vlasov-Klein-Gordon system.The main results of this article are also stated in this chapter.The second chapter discusses the Vlasov-Poisson system with radiation damping D_f^[2]?t?.First,by virtue of the generalized kinetic energy?i.e.velocity-space moment of order two?we establish a new conservation law and get the priori bound for velocity-space moment of order two and potential energy.Then,based on these,we prove the global ex-istence and uniqueness of classical solutions and establish the asymptotic behaviors of the solutions.Finally,with the condition of the boundedness of velocity-space moment of order two but may infinite kinetic energy,we prove the global existence of weak solutions.The third chapter,we discuss the propagation of velocity and space moments of weak solutions for this system,and establish some propagation results for low moments of weak solutions.The fourth chapter considers the Vlasov-Poisson system with radiation damping D_f^[3]?t?.The classical solution of this system in three dimensional for general initial is open.In this chapter,We prove that a small perturbation of initial datum for a global classical solution verifying certain decay conditions also launches a global solution which has sharp decay.In addition,we prove that a quasineutral initial datum launches a global classical solution.Chapter 5 and chapter 6 study the relativistic Vlasov-Klein-Gordon system.So far existence of global weak solutions in three or two dimensions has been established only for small initial data[70,79].In chapter 5,we obtain global existence of weak solutions for general initial data to the two dimensional relativistic Vlasov-Klein-Gordon system.In chapter 6,the relativistic Vlasov-Klein-Gordon system in one dimension is investi-gated.Without any assumption of continuity or compact support of any initial particle density f0,by applying the iteration method we prove the existence and uniqueness of the mild solutions.Specifically,we establish finite speed propagation result of the mild solution.