Diffusive transport of partially quantized particles: existence, uniqueness and long time behaviour

A selfconsistent model for charged particles, accounting for quantum confinement, diffusive transport and electrostatic interaction is considered. The electrostatic potential is a solution of a three dimensional Poisson equation with the particle density as the source term. This density is the product of a two dimensional surface density and that of a one dimensional mixed quantum state. The surface density is the solution of a drift-diffusion equation with an effective surface potential deduced from the fully three dimensional one and which involves the diagonalization of a one dimensional Schrödinger operator. The overall problem is viewed as a two dimensional drift-diffusion equation coupled to a Schrödinger-Poisson system. The latter is proven to be well posed by a convex minimization technique. A relative entropy and an a priori L2 estimate provide enough bounds to prove existence and uniqueness of a global in time solution. In the case of thermodynamic equilibrium boundary data, a unique stationary solution is proven to exist. The relative entropy allows to prove the convergence of the transient solution towards it as time grows to infinity. Finally, the low order approximation of the relative entropy is used to prove that this convergence is exponential in time.