Existence Of A Solution For Vlasov-Poisson System With Magnetic Confinement Effect

As we all know,the purpose of many technological applications(such as fusion research)is to control plasma through a magnetic field in an ideal way.When the plasma temperature is not very high,it can be described by the Vlasov-Poisson equation.This article discusses the time evolution of a single positive ion in an infinite cylinder,in which the total mass of the plasma is limited,but the phase space density function is infinite with respect to the velocity pillar.The plasma is constrained by an external magnetic field and an external potential,both of which tend to infinity at the edge of an infinite cylinder.The main purpose of this paper is to study the existence and uniqueness of the solution of the Vlasov-Poisson system with magnetic confinement effect in the cylinder.The first chapter introduces the research background of the classic Vlasov-Poisson system and the research status of the external magnetic field Vlasov-Poisson system,and gives the main research results of this paper.The second chapter constructs the solution of the approximation system by truncating the initial density along the velocity direction,and obtains the existence and uniqueness of the approximation system solution and the relevant prior estimation of the approximation system.The third chapter obtains the existence and uniqueness of the solution of the original system by taking the limit of the solution of the approaching system,and proves that the pillars of the plasma along the space direction are tightly contained in the cylinder.Therefore,the magnetic confinement effect is realized in this infinite cylinder.