On global weak solutions of the Vlasov–Poisson equations with external magnetic field

We consider the first mixed problem for the system of Vlasov–Poisson equations with a given external magnetic field in a bounded domain. This problem describes the kinetics of high-temperature plasma in controlled thermonuclear fusion plants and is considered with respect to unknown functions: electric field potential, distribution functions of positively charged ions and electrons. Additionally, we assumed that the distribution functions of charged particles satisfy the condition of mirror reflection from the boundary of the domain under consideration. We prove the existence of global weak solutions of such a problem.