Uniqueness Of The Solution To The Vlasov–poisson System With A Point Charge

Plasma physics is one of the important research topics in astrophysics,life science,atmospheric physics and other disciplines.Since it is impossible to know the specific position of particle orbit,the research on plasma physics cannot use Newton's laws to study its orbital motion behavior,and the statistical physics method is usually used to describe its motion law.The Vlasov-Poisson system can be used to describe the evolution law of the plasma microdensity function according to the Vlasov equation under the action of the self-consistent field generated by Poisson equation.Compared with the classical Vlasov-Poisson system,the plasma particles in the Vlasov-Poisson system with a point charge are not only affected by the self-consistent field generated by the system itself,but also by the coulomb force generated by the point charge,which makes the research more difficult.There are two directions in the study of the Vlasov-Poisson equation with point charge:repulsion field and gravitational field.There are two directions to study the Vlasov-Poisson equation with a point charge:repulsion field and gravitational field,which mean that the plasma and point charge have the same charge and opposite charge respectively.In this paper,the Vlasov-Poisson system with a point charge in n(n=2,3)dimension under the repulsion field is studied.The uniqueness of the solution of the system is proved,and the initial microscopic and macroscopic density functions satisfying the uniqueness condition are obtained.Firstly,by introducing a new distance function,we prove that when the Lp norm of the plasma macroscopic density grow at most linearly with respect to p,the system has at most one weak solution on[0,T],which is Theorem 2.1.Then,in the Theorem 2.2,by using velocity moment control ??(t,x)?Lp,prove the uniqueness conditions of the Theorem 2.1 can be satisfied,when the moments in m higher than 1 of initial data can be controlled by(Cm)m/n.Finally,by using the property that the growth of ?(ln-|x|)k dx can be controlled by(Ck)k,it is proved that the assumption of the initial value in the Theorem 2.2 can be satisfied when the initial macro density function satisfies certain logarithmic growth,which is Theorem 2.3.