Flow characteristics in a crowded transport model

The aim of this paper is to discuss the appropriate modelling of in- and outflow boundary conditions for nonlinear drift-diffusion models for the transport of particles including size exclusion and their effect on the behaviour of solutions. We use a derivation from a microscopic asymmetric exclusion process and its extension to particles entering or leaving on the boundaries. This leads to specific Robin-type boundary conditions for inflow and outflow, respectively. For the stationary equation we prove the existence of solutions in a suitable set-up. Moreover, we investigate the flow characteristics for a small diffusion parameter \varepsilon , which yields the occurrence of a maximal current phase in addition to well-known one-sided boundary layer effects for linear drift-diffusion problems. In a 1D set-up we provide rigorous estimates in terms of ɛ, which confirm three different phases. Finally, we derive a numerical approach to solve the problem also in multiple dimensions.