# American Institute of Mathematical Sciences

December  2019, 24(12): 6511-6531. doi: 10.3934/dcdsb.2019151

## Multi-scale modeling of processes in porous media - coupling reaction-diffusion processes in the solid and the fluid phase and on the separating interfaces

 Hasselt University, Faculty of Sciences, Campus Diepenbeek, Agoralaan Gebouw D, Diepenbeek 3590, Belgium

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

Received  December 2018 Revised  January 2019 Published  December 2019 Early access  July 2019

The aim of this paper is the derivation of general two-scale compactness results for coupled bulk-surface problems. Such results are needed for example for the homogenization of elliptic and parabolic equations with boundary conditions of second order in periodically perforated domains. We are dealing with Sobolev functions with more regular traces on the oscillating boundary, in the case when the norm of the traces and their surface gradients are of the same order. In this case, the two-scale convergence results for the traces and their gradients have a similar structure as for perforated domains, and we show the relation between the two-scale limits of the bulk-functions and their traces. Additionally, we apply our results to a reaction diffusion problem of elliptic type with a Wentzell-boundary condition in a multi-component domain.

Citation: Markus Gahn. Multi-scale modeling of processes in porous media - coupling reaction-diffusion processes in the solid and the fluid phase and on the separating interfaces. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6511-6531. doi: 10.3934/dcdsb.2019151
##### References:
 [1] E. Acerbi, V. Chiadò, G. D. Maso and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Analysis, theory, Methods & Applications, 18 (1992), 481-496.  doi: 10.1016/0362-546X(92)90015-7. [2] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084. [3] G. Allaire, A. Damlamian and U. Hornung, Two-scale convergence on periodic surfaces and applications, in Proceedings of the International Conference on Mathematical Modelling of Flow Through Porous Media, A. Bourgeat et al., eds., World Scientific, Singapore, 15–25. [4] G. Allaire and H. Hutridurga, Homogenization of reactive flows in porous media and competition between bulk and surface diffusion, IMA Journal of Applied Mathematics, 77 (2012), 788-215.  doi: 10.1093/imamat/hxs049. [5] M. Amar and R. Gianni, Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices, Discrete Cont. Dyn. Sys. B, 23 (2018), 1739-1756.  doi: 10.3934/dcdsb.2018078. [6] A. Bendali and K. Lemrabet, The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation, SIAM J. Appl. Math., 56 (2010), 1664-1693.  doi: 10.1137/S0036139995281822. [7] J.-M. E. Bernard, Density results in Sobolev spaces whose elements vanish on a part of the boundary, Chin. Ann. Math., 32B (2011), 823-846.  doi: 10.1007/s11401-011-0682-z. [8] V. Bonnaillie-Noël, M. Dambrine, F. Héau and G. Vial, On generalized Ventcel's boundary conditions for Laplace operator in a bounded domain, SIAM J. Math. Anal., 42 (2010), 931-945.  doi: 10.1137/090756521. [9] 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. [10] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 3 Spectral Theory and Applications, Springer, 1990. [11] H. Douanla, Two-scale convergence of periodic elliptic spectral problems with indefinite density function in perforated domains, Asymptotic Analysis, 81 (2013), 251-272. [12] M. Gahn, M. 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. [13] G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480. [14] I. Graf and M. A. Peter, A convergence result for the periodic unfolding method related to fast diffusion on manifolds, C. R. Acad. Sci. Paris, Ser. I, 352 (2014), 485-490.  doi: 10.1016/j.crma.2014.03.002. [15] H. Hutridurga, Homogenization of Complex Flow in Porous Media and Applications, PhD thesis, École Polytechnique, 2013. [16] M. Liero, Passing from bulk to bulk-surface evolution in the allen-cahn equation, Nonlinear Differential Equations and Applications NoDEA, 20 (2012), 919-942.  doi: 10.1007/s00030-012-0189-7. [17] J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Academia, Praha & Masson, 1967. [18] M. Neuss-Radu, Some extensions of two-scale convergence, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 899-904. [19] 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. [20] J. Peetre, Another approach to elliptic boundary problems, Communications on Pure and Applied Mathematics, 14 (1961), 711-731.  doi: 10.1002/cpa.3160140404. [21] 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. [22] J. Sokolowski and J.-P. Zolesio, Introduction to Shape Optimization, Springer-Verlag, 1992. [23] R. S. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, Journal of Functional Analysis, 52 (1983), 48-79.  doi: 10.1016/0022-1236(83)90090-3. [24] H. Triebel, Theory of Function Spaces II, Birkhäuser, 1992. [25] A. D. Venttsel, On boundary conditions for multidimensional diffusion processes, Theor. Probability Appl., 4 (1959), 164-177.  doi: 10.1137/1104014. [26] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9781139171755.

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##### References:
 [1] E. Acerbi, V. Chiadò, G. D. Maso and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Analysis, theory, Methods & Applications, 18 (1992), 481-496.  doi: 10.1016/0362-546X(92)90015-7. [2] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084. [3] G. Allaire, A. Damlamian and U. Hornung, Two-scale convergence on periodic surfaces and applications, in Proceedings of the International Conference on Mathematical Modelling of Flow Through Porous Media, A. Bourgeat et al., eds., World Scientific, Singapore, 15–25. [4] G. Allaire and H. Hutridurga, Homogenization of reactive flows in porous media and competition between bulk and surface diffusion, IMA Journal of Applied Mathematics, 77 (2012), 788-215.  doi: 10.1093/imamat/hxs049. [5] M. Amar and R. Gianni, Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices, Discrete Cont. Dyn. Sys. B, 23 (2018), 1739-1756.  doi: 10.3934/dcdsb.2018078. [6] A. Bendali and K. Lemrabet, The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation, SIAM J. Appl. Math., 56 (2010), 1664-1693.  doi: 10.1137/S0036139995281822. [7] J.-M. E. Bernard, Density results in Sobolev spaces whose elements vanish on a part of the boundary, Chin. Ann. Math., 32B (2011), 823-846.  doi: 10.1007/s11401-011-0682-z. [8] V. Bonnaillie-Noël, M. Dambrine, F. Héau and G. Vial, On generalized Ventcel's boundary conditions for Laplace operator in a bounded domain, SIAM J. Math. Anal., 42 (2010), 931-945.  doi: 10.1137/090756521. [9] 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. [10] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 3 Spectral Theory and Applications, Springer, 1990. [11] H. Douanla, Two-scale convergence of periodic elliptic spectral problems with indefinite density function in perforated domains, Asymptotic Analysis, 81 (2013), 251-272. [12] M. Gahn, M. 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. [13] G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480. [14] I. Graf and M. A. Peter, A convergence result for the periodic unfolding method related to fast diffusion on manifolds, C. R. Acad. Sci. Paris, Ser. I, 352 (2014), 485-490.  doi: 10.1016/j.crma.2014.03.002. [15] H. Hutridurga, Homogenization of Complex Flow in Porous Media and Applications, PhD thesis, École Polytechnique, 2013. [16] M. Liero, Passing from bulk to bulk-surface evolution in the allen-cahn equation, Nonlinear Differential Equations and Applications NoDEA, 20 (2012), 919-942.  doi: 10.1007/s00030-012-0189-7. [17] J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Academia, Praha & Masson, 1967. [18] M. Neuss-Radu, Some extensions of two-scale convergence, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 899-904. [19] 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. [20] J. Peetre, Another approach to elliptic boundary problems, Communications on Pure and Applied Mathematics, 14 (1961), 711-731.  doi: 10.1002/cpa.3160140404. [21] 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. [22] J. Sokolowski and J.-P. Zolesio, Introduction to Shape Optimization, Springer-Verlag, 1992. [23] R. S. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, Journal of Functional Analysis, 52 (1983), 48-79.  doi: 10.1016/0022-1236(83)90090-3. [24] H. Triebel, Theory of Function Spaces II, Birkhäuser, 1992. [25] A. D. Venttsel, On boundary conditions for multidimensional diffusion processes, Theor. Probability Appl., 4 (1959), 164-177.  doi: 10.1137/1104014. [26] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9781139171755.
Microscopic domain for $\epsilon = \frac14$ and $\Omega = (0,1)^2$, and $Y_2$ is strictly included in $Y$
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