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 is established in terms of the norm of the initial density distribution functions. Sufficient conditions are found for the supports (with respect to the space variables) of the density distribution functions to lie at a prescribed distance from the boundary, which corresponds to plasma confinement in a fusion reactor. © 2025 Russian Academy of Sciences, Steklov Mathematical Institute of RAS.