Analysis of the Relativistic Vlasov-maxwell Model in an Interval

The dynamics of a collisionless electron gas under the influence of the selfconsistent electromagnetic field is studied in an interval, where electrons are emitted at one end of the interval and absorbed at the other. The electron distribution f depends on one space variable x and two dimensional velocity variables (v1, v2), whereas the electromagnetic field satisfies the Maxwell system. It is shown that the electromagnetic field is smooth enough to define the trajectories of particles. On the other hand, due to the absorbing boundary condition, f is in general not smooth, and only has bounded total variations. Such BV regularity is sufficient to guarantee the uniqueness of the nonlinear electron dynamics. Finally, numerical simulations are performed to demonstrate the formation of a singularity from the boundary for the electron distribution.