Stationary solutions of the Vlasov-Poisson system and plasma confinement in a tokamak.

In this lecture we consider the Vlasov-Poisson system for two-component high-temperature plasma with external magnetic field in a tokamak. If a considerable part of particles reaches the boundary, this can lead to destruction of the reactor. Therefore, it is necessary to provide plasma confinement at some distance from the vacuum container wall. From the point of view of differential equations this means that one has to prove the existence of solutions of the Vlasov-Poisson system with external magnetic field for which the supports of density distribution functions do not intersect with the boundary. We prove the existence of stationary solutions of the Vlasov-Poisson system in a three-dimensional torus with compactly supported density distribution functions.