Remark on a magnetically confined plasma with infinite charge

We consider a one-species plasma moving in an infinite cylinder in ${\mathbb{R}}^3$, in which it is confined by means of a magnetic field diverging on the walls of the cylinder. In a recent paper \cite{CCM2} we have supposed that the plasma satisfies the Vlasov equation with a Yukawa mutual interaction (i.e. Coulomb at short distance and exponentially decreasing at infinity). Assuming that initially the particles have bounded velocities and are distributed according to a bounded density without any hypothesis on its decreasing at infinity, we have proved the global in time existence and uniqueness of the time evolution of the plasma and its confinement. In the present paper we extend this result to a Coulomb interaction, making on the initial density some assumption of slight decreasing on average at infinity, which however does not imply that the density belongs to any $L^p$ space. The proof is similar, but slightly simpler.