Vlasov-Poisson equations for a two-component plasma in a~homogeneous magnetic field (Q2875028)

The paper deals with the mixed problem for the Vlasov-Poisson equations in an infinite cylinder with the Dirichlet boundary condition. This mathematical model describes the evolution of the density distribution of ions and electrons in a high temperature plasma under an external magnetic field. The author obtains the global classical solution, such that the supports of the charged-particle density distributions lie at some distance from the cylindrical surface. From the physical point of view, this condition corresponds to the situation in the thermonuclear fusion reactor, when the charged particles do not reach the walls of the vacuum chamber.NEWLINENEWLINEThe author proves the existence of a stationary solution of the Vlasov-Poisson equations with the charged-particle density distributions, supported in a strictly interior cylinder, and then constructs a unique classical solution in a neighbourhood of this stationary solution. The main ingredients of the proofs are Hölder estimates and the Banach fixed point theorem.NEWLINENEWLINEThe author also gives a survey of the extensive literature, concerning initial value problems and mixed problems for the Vlasov equations and their physical applications in the introduction. In conclusion, he provides the generalization of the main result to some class of abstract Vlasov equations, and gives a list of open problems.