# 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  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 & Continuous Dynamical Systems - B, 2019, 24 (12) : 6511-6531. doi: 10.3934/dcdsb.2019151
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##### References:
Microscopic domain for $\epsilon = \frac14$ and $\Omega = (0,1)^2$, and $Y_2$ is strictly included in $Y$
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