Derivation of effective transmission conditions for domains separated by a membrane for different scaling of membrane diffusivity

Markus Gahn; Maria Neuss-Radu; Peter Knabner

doi:10.3934/dcdss.2017039

Article Contents

2017, Volume 10, Issue 4: 773-797. Doi: 10.3934/dcdss.2017039

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Derivation of effective transmission conditions for domains separated by a membrane for different scaling of membrane diffusivity

Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße 11,91058 Erlangen, Germany

^* Corresponding author: Markus Gahn
^* Corresponding author: Markus Gahn

Received: April 30, 2016

Accepted: June 30, 2016

Published: July 2017

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Abstract

We consider a system of non-linear reaction-diffusion equations in a domain consisting of two bulk regions separated by a thin layer with periodic structure. The thickness of the layer is of order $\epsilon$, and the equations inside the layer depend on the parameter $\epsilon$ and an additional parameter $\gamma \in [-1,1)$, which describes the size of the diffusion in the layer. We derive effective models for the limit $\epsilon \to 0 $, when the layer reduces to an interface $\Sigma$ between the two bulk domains. The effective solution is continuous across $\Sigma$ for all $\gamma \in [-1,1)$. For $\gamma \in (-1,1)$, the jump in the normal flux is given by a non-linear ordinary differential equation on $\Sigma$. In the critical case $\gamma = -1$, a dynamic transmission condition of Wentzell-type arises at the interface $\Sigma$.

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Mathematics Subject Classification: Primary: 35K57, 35B27, 80M40.

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Figure 1. The microscopic domain containing the thin layer $\Omega _\epsilon^ M$ with periodic structure. The heterogeneous structure of the membrane is modeled by the diffusion-coefficient $D^M$. In biology such a layer is e. g., the stratum corneum which consists of flattened cells (corneocytes) surrounded by lipid components

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