Effective interface conditions for processes through thin heterogeneous layers with nonlinear transmission at the microscopic bulk-layer interface

Markus Gahn; Maria Neuss-Radu; Peter Knabner

doi:10.3934/nhm.2018028

Article Contents

2018, Volume 13, Issue 4: 609-640. Doi: 10.3934/nhm.2018028

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Effective interface conditions for processes through thin heterogeneous layers with nonlinear transmission at the microscopic bulk-layer interface

1.
Center for Modelling and Simulation in the Biosciences (BIOMS), Interdisciplinary Center for Scientific Computing (IWR), University of Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
2.
Applied Mathematics I, Department Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany

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

Received: January 2018

Revised: July 2018

Published: November 2018

The work of the first author was supported by the Center for Modelling and Simulation in the Biosciences (BIOMS) at the University of Heidelberg.

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Abstract

In this paper, we consider a system of reaction-diffusion equations in a domain consisting of two bulk regions separated by a thin layer with thickness of order $ε$ and a periodic heterogeneous structure. The equations inside the layer depend on $ε$ and the diffusivity inside the layer on an additional parameter $γ ∈ [-1, 1]$. On the bulk-layer interface, we assume a nonlinear Neumann-transmission condition depending on the solutions on both sides of the interface. For $\epsilon \to0 $, when the thin layer reduces to an interface $Σ$ between two bulk domains, we rigorously derive macroscopic models with effective conditions across the interface $Σ$. The crucial part is to pass to the limit in the nonlinear terms, especially for the traces on the interface between the different compartments. For this purpose, we use the method of two-scale convergence for thin heterogeneous layers, and a Kolmogorov-type compactness result for Banach valued functions, applied to the unfolded sequence in the thin layer.

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Mathematics Subject Classification: 35B27, 35K57, 35K61, 80M40.

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Figure 1. The microscopic domain containing the thin layer $\Omega_\epsilon^M$ with periodic structure for $n = 2$. The heterogeneous structure of the membrane is modeled by the diffusion coefficient $D^M$

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