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; Markus Gahn; Maria Neuss-Radu; Peter Knabner

doi:10.3934/nhm.2018028

Networks and Heterogeneous Media

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

Received: 01 January 2018 Revised: 01 July 2018
35B27, 35K57, 35K61, 80M40

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.
Citation: Markus Gahn, Maria Neuss-Radu, Peter Knabner. Effective interface conditions for processes through thin heterogeneous layers with nonlinear transmission at the microscopic bulk-layer interface[J]. Networks and Heterogeneous Media, 2018, 13(4): 609-640. doi: 10.3934/nhm.2018028

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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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