Flow Characteristics in a Crowded Transport Model

Abstract

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 setup. Moreover, we investigate the flow characteristics for small diffusion, which yields the occurence of a maximal current phase in addition to well-known one-sided boundary layer effects for linear drift-diffusion problems. In a one-dimensional setup we provide rigorous estimates in terms of $\epsilon$, which confirm three different phases. Finally, we derive a numerical approach to solve the problem also in multiple dimensions. This provides further insight and allows for the investigation of more complicated geometric setups.

References in corpus (3)

• The boundedness-by-entropy principle for cross-diffusion systems
• Crossing pedestrian traffic flows,diagonal stripe pattern, and chevron effect
• Stepping and crowding of molecular motors: statistical kinetics from an exclusion process perspective

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