Flow characteristics in a crowded transport model. (30th September 2016)

• Record Type: Journal Article
• Authors: Burger, Martin; Pietschmann, Jan-Frederik
• Is Part Of: Nonlinearity. Volume 29:Number 11(2016:Nov.)
• Page Start: 3528
• Page End: 3550
• Publication Date: 2016-09-30
• Subjects: flow characteristics -- crowded transport -- partial differential equations; 35K20 -- 35K55 -- 82C20 -- 82C31 -- 92D99; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/0951-7715/29/11/3528
• Languages: English
• ISSNs: 0951-7715

Abstract

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 set-up. Moreover, we investigate the flow characteristics for a small diffusion parameter ε, 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 ε, which confirm three different phases. Finally, we derive a numerical approach to solve the problem also in multiple dimensions.