拥挤运输模型中的流动特性
Flow Characteristics in a Crowded Transport Model
摘要：
本文的目的是讨论适当的建模和流出边界条件的非线性漂移扩散模型的颗粒传输，包括尺寸排阻及其对行为的影响的解决方案。我们使用一个派生的微观不对称排它过程及其扩展到粒子进入或离开的边界。这导致流入和流出的特定罗宾型边界条件，分别。对于定常方程，我们证明了在适当的设置的解决方案的存在性。此外，我们研究了小扩散的流动特性，除了众所周知的单侧边界层效应的线性漂移-扩散问题，产生了最大电流相的发生。在一个一维的设置，我们提供严格的估计方面的$ε$，确认三个不同的阶段。最后，我们推导出一个数值方法来解决这个问题，也在多个维度。这提供了进一步的见解，并允许更复杂的几何设置的调查。
英文摘要：
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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