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arXiv · 1502.02715

Flow Characteristics in a Crowded Transport Model

Martin Burger·Jan-Frederik Pietschmann
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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 $ε$, 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.

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Martin Burger, Jan-Frederik Pietschmann. 2015-02-09. Flow Characteristics in a Crowded Transport Model. https://doi.org/10.1088/0951-7715%2F29%2F11%2F3528

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