December  2018, 13(4): 609-640. doi: 10.3934/nhm.2018028

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

Received  January 2018 Revised  July 2018 Published  November 2018

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

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. Networks & Heterogeneous Media, 2018, 13 (4) : 609-640. doi: 10.3934/nhm.2018028
References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.  Google Scholar

[2]

T. ArbogastJ. Douglas and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 27 (1990), 823-836.  doi: 10.1137/0521046.  Google Scholar

[3]

A. BourgeatO. Gipouloux and E. Marušić-Paloka, Modelling of an underground waste disposal site by upscaling, Math. Meth. Appl. Sci., 27 (2004), 381-403.  doi: 10.1002/mma.459.  Google Scholar

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A. Bourgeat and E. Marušić-Paloka, A homogenized model of an underground waste repository including a disturbed zone, Multiscale Model. Simul., 3 (2005), 918-939.  doi: 10.1137/040605424.  Google Scholar

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H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011.  Google Scholar

[6]

D. CioranescuA. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620.  doi: 10.1137/080713148.  Google Scholar

[7]

D. CioranescuA. DamlamianG. Griso and D. Onofrei, The periodic unfolding method for perforated domains and Neumann sieve models, J. Math. Pures Appl., 89 (2008), 248-277.  doi: 10.1016/j.matpur.2007.12.008.  Google Scholar

[8]

M. Gahn and M. Neuss-Radu, A characterization of relatively compact sets in Lp(Ω, B), Stud. Univ. Babeş-Bolyai Math., 61 (2016), 279-290.   Google Scholar

[9]

M. GahnM. Neuss-Radu and P. Knabner, Homogenization of reaction-diffusion processes in a two-component porous medium with nonlinear flux conditions at the interface, SIAM Journal on Applied Mathematics, 76 (2016), 1819-1843.  doi: 10.1137/15M1018484.  Google Scholar

[10]

M. GahnM. Neuss-Radu and P. Knabner, Derivation of effective transmission conditions for domains separated by a membrane for different scaling of membrane diffusivity, Discrete & Continuous Dynamical Systems-Series S, 10 (2017), 773-797.  doi: 10.3934/dcdss.2017039.  Google Scholar

[11]

G. GrisoA. Migunova and J. Orlik, Homogenization via unfolding in periodic layer with contact, Asymptotic Analysis, 99 (2016), 23-52.  doi: 10.3233/ASY-161374.  Google Scholar

[12]

G. GrisoA. Migunova and J. Orlik, Asymptotic analysis for domains separated by a thin layer made of periodic vertical beams, Journal of Elasticity, 128 (2017), 291-331.  doi: 10.1007/s10659-017-9628-3.  Google Scholar

[13]

A. A. Moussa and L. Zlaïji, Homogenization of non-linear variational problems with thin inclusions, Math. J. Okayama Univ., 54 (2012), 97-131.   Google Scholar

[14]

M. Neuss-Radu, Mathematical modelling and multi-scale analysis of transport processes through membranes (habilitation thesis), University of Heidelberg, 2017. Google Scholar

[15]

M. Neuss-Radu and W. Jäger, Effective transmission conditions for reaction-diffusion processes in domains separated by an interface, SIAM J. Math. Anal., 39 (2007), 687-720.  doi: 10.1137/060665452.  Google Scholar

[16]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.  doi: 10.1137/0520043.  Google Scholar

[17]

I. S. PopJ. Bogers and K. Kumar, Analysis and upscaling of a reactive transport model in fractured porous media with nonlinear transmission condition, Vietnam Journal of Mathematics, 45 (2017), 77-102.  doi: 10.1007/s10013-016-0198-7.  Google Scholar

[18]

C. Vogt, A Homogenization Theorem Leading to a Volterra Integro-Differential Equation for Permeation Chromotography, SFB 123, University of Heidelberg, Preprint 155 and Diploma-thesis, 1982. Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.  Google Scholar

[2]

T. ArbogastJ. Douglas and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 27 (1990), 823-836.  doi: 10.1137/0521046.  Google Scholar

[3]

A. BourgeatO. Gipouloux and E. Marušić-Paloka, Modelling of an underground waste disposal site by upscaling, Math. Meth. Appl. Sci., 27 (2004), 381-403.  doi: 10.1002/mma.459.  Google Scholar

[4]

A. Bourgeat and E. Marušić-Paloka, A homogenized model of an underground waste repository including a disturbed zone, Multiscale Model. Simul., 3 (2005), 918-939.  doi: 10.1137/040605424.  Google Scholar

[5]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011.  Google Scholar

[6]

D. CioranescuA. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620.  doi: 10.1137/080713148.  Google Scholar

[7]

D. CioranescuA. DamlamianG. Griso and D. Onofrei, The periodic unfolding method for perforated domains and Neumann sieve models, J. Math. Pures Appl., 89 (2008), 248-277.  doi: 10.1016/j.matpur.2007.12.008.  Google Scholar

[8]

M. Gahn and M. Neuss-Radu, A characterization of relatively compact sets in Lp(Ω, B), Stud. Univ. Babeş-Bolyai Math., 61 (2016), 279-290.   Google Scholar

[9]

M. GahnM. Neuss-Radu and P. Knabner, Homogenization of reaction-diffusion processes in a two-component porous medium with nonlinear flux conditions at the interface, SIAM Journal on Applied Mathematics, 76 (2016), 1819-1843.  doi: 10.1137/15M1018484.  Google Scholar

[10]

M. GahnM. Neuss-Radu and P. Knabner, Derivation of effective transmission conditions for domains separated by a membrane for different scaling of membrane diffusivity, Discrete & Continuous Dynamical Systems-Series S, 10 (2017), 773-797.  doi: 10.3934/dcdss.2017039.  Google Scholar

[11]

G. GrisoA. Migunova and J. Orlik, Homogenization via unfolding in periodic layer with contact, Asymptotic Analysis, 99 (2016), 23-52.  doi: 10.3233/ASY-161374.  Google Scholar

[12]

G. GrisoA. Migunova and J. Orlik, Asymptotic analysis for domains separated by a thin layer made of periodic vertical beams, Journal of Elasticity, 128 (2017), 291-331.  doi: 10.1007/s10659-017-9628-3.  Google Scholar

[13]

A. A. Moussa and L. Zlaïji, Homogenization of non-linear variational problems with thin inclusions, Math. J. Okayama Univ., 54 (2012), 97-131.   Google Scholar

[14]

M. Neuss-Radu, Mathematical modelling and multi-scale analysis of transport processes through membranes (habilitation thesis), University of Heidelberg, 2017. Google Scholar

[15]

M. Neuss-Radu and W. Jäger, Effective transmission conditions for reaction-diffusion processes in domains separated by an interface, SIAM J. Math. Anal., 39 (2007), 687-720.  doi: 10.1137/060665452.  Google Scholar

[16]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.  doi: 10.1137/0520043.  Google Scholar

[17]

I. S. PopJ. Bogers and K. Kumar, Analysis and upscaling of a reactive transport model in fractured porous media with nonlinear transmission condition, Vietnam Journal of Mathematics, 45 (2017), 77-102.  doi: 10.1007/s10013-016-0198-7.  Google Scholar

[18]

C. Vogt, A Homogenization Theorem Leading to a Volterra Integro-Differential Equation for Permeation Chromotography, SFB 123, University of Heidelberg, Preprint 155 and Diploma-thesis, 1982. Google Scholar

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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