A priori estimate of solutions to the first mixed problem for Vlasov–Poisson system and plasma confinement

The first mixed problem is considered for the Vlasov–Poisson system of equations with an external magnetic field with respect to the potential of a self-consistent electric field and the distribution functions of the densities of charged particles. An a priori estimate for the norm of the gradient of the potential of the electric field in terms of the norm of the initial functions of the density distribution is established. Sufficient conditions are found for the supports (in the space variables) of the functions of the density distribution to lie at a prescribed distance from the boundary, which corresponds to plasma confinement in a fusion reactor.