August  2017, 10(4): 773-797. doi: 10.3934/dcdss.2017039

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

Received  May 2016 Accepted  July 2016 Published  April 2017

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 $\Sigma$.

Citation: Markus Gahn, Maria Neuss-Radu, Peter Knabner. Derivation of effective transmission conditions for domains separated by a membrane for different scaling of membrane diffusivity. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 773-797. doi: 10.3934/dcdss.2017039
References:
[1]

T. ArbogastJ. Douglas and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 21 (1990), 823-836.  doi: 10.1137/0521046.

[2]

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.

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011. doi: 10.1007/978-0-387-70914-7.

[4]

J. R. Cannon and G. H. Meyer, On diffusion in a fractured medium, SIAM J. Appl. Math., 20 (1971), 434-448.  doi: 10.1137/0120047.

[5]

D. CioranescuA. Damlamian and G. Griso, Periodic unfolding and homogenization, C. R. Acad. Sci. Paris Sér. 1, 335 (2002), 99-104.  doi: 10.1016/S1631-073X(02)02429-9.

[6]

P. Donato and A. Piatnitski, On the effective interfacial resistance through rough surfaces, Commun. Pure Appl. Anal., 9 (2010), 1295-1310. 

[7]

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. 

[8]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer Monographs in Mathematics, 2011.

[9]

G. GeymonatS. HendiliF. Krasucki and M. Vidrascu, Matched asymptotic expansion method for an homogenized interface model, Math. Models, Methods Appl. Sci., 24 (2014), 573-597.  doi: 10.1142/S0218202513500607.

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983.

[11]

M. Liero, Passing from bulk to bulk-surface evolution in the allen-cahn equation, Nonlinear Differential Equations and Applications NoDEA, 20 (2013), 919-942.  doi: 10.1007/s00030-012-0189-7.

[12]

S. Marušić and E. Marušić-Paloka, Two-scale convergence for thin domains and its applications to some lower-dimensional mmodel in fluid mechanics, Asymptotic Analysis, 23 (2000), 23-58. 

[13]

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

[14]

M. Neuss-Radu, Mathematical Modelling and Multi-Scale Analysis of Transport Processes Through Membranes, Habilitation Thesis, University of Heidelberg, 2017.

[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.

[16]

M. Neuss-Radu and W. J. S. Ludwig, Multiscale analysis and simulation of a reaction-diffusion problem with transmission conditions, Nonlinear Analysis: Real World Applications, 11 (2010), 4572-4585.  doi: 10.1016/j.nonrwa.2008.11.024.

[17]

M. A. Peter and M. Böhm, Different choices of scaling in homogenization of diffusion and interfacial exchange in a porous medium, Math. Meth. Appl. Sci., 31 (2008), 1257-1282.  doi: 10.1002/mma.966.

[18]

M. Shinbrot, Water waves over periodic bottoms in three dimensions, J. Inst. Maths. Applics., 25 (1980), 367-385.  doi: 10.1093/imamat/25.4.367.

[19]

A. D. Venttsel, On boundary conditions for multidimensional diffusion processes, Teor. Veroyatnost. i Primenen., 4 (1959), 172-185.  doi: 10.1137/1104014.

[20]

J. Wloka, Partielle Differentialgleichungen, B. G. Teubner, 1982.

show all references

References:
[1]

T. ArbogastJ. Douglas and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 21 (1990), 823-836.  doi: 10.1137/0521046.

[2]

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.

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011. doi: 10.1007/978-0-387-70914-7.

[4]

J. R. Cannon and G. H. Meyer, On diffusion in a fractured medium, SIAM J. Appl. Math., 20 (1971), 434-448.  doi: 10.1137/0120047.

[5]

D. CioranescuA. Damlamian and G. Griso, Periodic unfolding and homogenization, C. R. Acad. Sci. Paris Sér. 1, 335 (2002), 99-104.  doi: 10.1016/S1631-073X(02)02429-9.

[6]

P. Donato and A. Piatnitski, On the effective interfacial resistance through rough surfaces, Commun. Pure Appl. Anal., 9 (2010), 1295-1310. 

[7]

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. 

[8]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer Monographs in Mathematics, 2011.

[9]

G. GeymonatS. HendiliF. Krasucki and M. Vidrascu, Matched asymptotic expansion method for an homogenized interface model, Math. Models, Methods Appl. Sci., 24 (2014), 573-597.  doi: 10.1142/S0218202513500607.

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983.

[11]

M. Liero, Passing from bulk to bulk-surface evolution in the allen-cahn equation, Nonlinear Differential Equations and Applications NoDEA, 20 (2013), 919-942.  doi: 10.1007/s00030-012-0189-7.

[12]

S. Marušić and E. Marušić-Paloka, Two-scale convergence for thin domains and its applications to some lower-dimensional mmodel in fluid mechanics, Asymptotic Analysis, 23 (2000), 23-58. 

[13]

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

[14]

M. Neuss-Radu, Mathematical Modelling and Multi-Scale Analysis of Transport Processes Through Membranes, Habilitation Thesis, University of Heidelberg, 2017.

[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.

[16]

M. Neuss-Radu and W. J. S. Ludwig, Multiscale analysis and simulation of a reaction-diffusion problem with transmission conditions, Nonlinear Analysis: Real World Applications, 11 (2010), 4572-4585.  doi: 10.1016/j.nonrwa.2008.11.024.

[17]

M. A. Peter and M. Böhm, Different choices of scaling in homogenization of diffusion and interfacial exchange in a porous medium, Math. Meth. Appl. Sci., 31 (2008), 1257-1282.  doi: 10.1002/mma.966.

[18]

M. Shinbrot, Water waves over periodic bottoms in three dimensions, J. Inst. Maths. Applics., 25 (1980), 367-385.  doi: 10.1093/imamat/25.4.367.

[19]

A. D. Venttsel, On boundary conditions for multidimensional diffusion processes, Teor. Veroyatnost. i Primenen., 4 (1959), 172-185.  doi: 10.1137/1104014.

[20]

J. Wloka, Partielle Differentialgleichungen, B. G. Teubner, 1982.

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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