Uniqueness And Existence Of Classical Solutions To Vlasov-Poisson System With Radiation Damping

The Vlasov-Poisson system is a kind of kinetic model describing collisionless particles which interact by a field which they generate together. It has important applications in many research fields, which include astrophysics, semiconductors and plasma and so on.In classical electrodynamics, a part of electromagnetic field which is generated by ac-celerated charged particles is external radiation field. The radiation field leads to loss of energy by the particles gradually, therefore we need add radiation damping term to Vlasov-Poisson system. In this paper, we discuss uniqueness and existence of classical solutions to the Vlasov-Poisson equations with radiation damping under the assumption that initial conditions f0±∈Cb1(R3×R3) satisfy polynomial decay at infinity. We prove existence of classical solutions by Schauder fixed point theorem. Firstly, we give conditions of the set S. With definitions of convex set and closed set as well as Arzela-Ascoli theorem, we obtain that the set S is a compact convex set. Secondly, we can get some properties of functions f±by Holder estimate of characteristics and related estimates, moreover we can assure param-eters of the set S, so we get that the mapping V is S to own mapping and continuous. At the same time the scope of the variable t is given, namely the existent range of classical solu-tions. Afterwards, thank to the Schauder fixed point theorem, we show existence of classical solutions. Finally, we prove uniqueness of classical solutions with the aid of constructing Gronwall inequality about classical solutions.Prior to this, Kunze and Rendall give existence of local solutions in2001with article about the Vlasov-Poisson system with radiation damping, but it is under the assumption that initial conditions have compact supports. In this paper, the initial conditions are weaker than Kunze and Rendall’s article, namely the initial conditions polynomially decay at infinity.