The Global Classical Solutions And Asymptotic Behaviors Of Three Classes Of Vlasov Type Equations

This paper investigates the global classical solutions and asymptotic behaviors of three classes of Vlasov type equations:non-relativistic Vlasov-Darwin system,Vlasov-Poisson system with radiation damping D[2](t)and D[3](t).In Chapter 1,we elaborated on the research background,research status and research method of three kinds of approximation systems of Relativistic Vlasov-Maxwell system:(Relativistic)Vlasov-Poisson system,(Relativistic)Vlasov-Darwin system and VlasovPoisson system with radiation damping.In particular,the research on Vlasov-Poisson system has come to complete results in relative terms,and its methods and contents have important significance for other Vlasov-type equations.The last section of this chapter we summary the main research contents and results of this paper.In Chapter 2,we investigate the non-relativistic Vlasov-Darwin system with generalized variables(VDG system).We derive some estimates of current density function j(t,x)in terms of the conservation of the total energy to overcome the difficulty.And we prove the global existence and uniqueness for small initial data and derive the decay estimates of the Darwin potentials.Based upon this result,we further show that solutions of VDG system converge to that of VP system in global time as the speed of light tends to infinity and the asymptotic rate is 1/c2.Moreover,we obtain the asymptotic behavior of the difference between the solutions of the two systems under the same small initial data.The Chapter 3 studies the same system as the Chapter 2.By proving a local stability result,we obtain a small perturbation result under the classical solution satisfying some decay estimates.And generalizing the quasi-spherical-symmetry case,we prove the existence and uniqueness of the classical solution of the system.In Chapter 4,we consider asymptotic behaviors of the Vlasov-Poisson system with radiation damping D[2](t).we deal with the difficulty of damping term by introducing parameter n and taking special value.For any smooth solution with compact support,we establish a sub-linear growth estimate of its velocity support by applying Lagrange approach.As a consequence,we derive some new estimates of the charge densities and the electrostatic field in this situation.In Chapter 5,we are concerned with global wellposedness of the Vlasov-Poisson system with radiation damping D[3](t).In the previous literature,the global existence of the classical solution has been proved in the case of small perturbation and quasi-neutral initial value.However,these results require that the initial data must have higher order regularity than the solutions themselves,causing loss of regularity.In this chapter,we prove that any C1 initial data defines a global C1 solution with better decay estimates of the radiation damping D[3]in both cases mentioned above.In addition,we extend the first result to a monopolar and spherically symmetric plasma and prove the propagation of quasi-sphericalsymmetry and quasi-neutral at the macroscopic level.In the last chapter,we summarize this paper and propose some problems to be solved.