Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior

A collisionless plasma is modeled by theVlasov-Poisson system in three space dimensions. A fixedbackground of positive charge, which is independent of time andspace, is assumed. The situation in which mobile negative ionsbalance the positive charge as |x|→∞ is considered.Hence, the total positive charge and the total negative charge areboth infinite. It is shown, in three spatial dimensions, thatsmooth solutions may be continued as long as the velocity supportremains finite. Also, in the case of spherical symmetry, a boundon velocity support is obtained and hence solutions exist globallyin time.