Skubachevskii, A. L. Vlasov-Poisson equations for a two-component plasma in a homogeneous magnetic field. (English. Russian original) Zbl 1302.35369 Russ. Math. Surv. 69, No. 2, 291 (2014); translation from Usp. Mat. Nauk 69, No. 2, 107-148 (2014). 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.The 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.The 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. Reviewer: Natalia Bondarenko (Saratov) Cited in 30 Documents MSC: 35Q83 Vlasov equations 76X05 Ionized gas flow in electromagnetic fields; plasmic flow 82D10 Statistical mechanics of plasmas Keywords:Vlasov-Poisson equations; mixed problem; classical solutions; homogeneous magnetic field × Cite Format Result Cite Review PDF Full Text: DOI