An analysis of the Wigner–Poisson problem with inflow boundary conditions

We present a study of the Wigner–Poisson problem in a bounded spatial domain with non-homogeneous and time-dependent “inflow” boundary conditions. This system of nonlinearly coupled equations is a mathematical model for quantum transport of charges in a semiconductor with external contacts. We prove well-posedness of the linearized n-dimensional problem as well as existence and uniqueness of a global-in-time, regular solution of the one-dimensional nonlinear problem.