Singular limit for reactive transport through a thin heterogeneous layer including a nonlinear diffusion coefficient

Abstract

<p style='text-indent:20px;'>Reactive transport processes in porous media including thin heterogeneous layers play an important role in many applications. In this paper, we investigate a reaction-diffusion problem with nonlinear diffusion in a domain consisting of two bulk domains which are separated by a thin layer with a periodic heterogeneous structure. The thickness of the layer, as well as the periodicity within the layer are of order <inline-formula><tex-math id="M1">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> is much smaller than the size of the bulk domains. For the singular limit <inline-formula><tex-math id="M3">\begin{document}$ \epsilon \to 0 $\end{document}</tex-math></inline-formula>, when the thin layer reduces to an interface, we rigorously derive a macroscopic model with effective interface conditions between the two bulk domains. Due to the oscillations within the layer, we have the combine dimension reduction techniques with methods from the homogenization theory. To cope with these difficulties, we make use of the two-scale convergence in thin heterogeneous layers. However, in our case the diffusion in the thin layer is low and depends nonlinearly on the concentration itself. The low diffusion leads to a two-scale limit depending on a macroscopic and a microscopic variable. Hence, weak compactness results based on standard <i>a priori</i> estimates are not enough to pass to the limit <inline-formula><tex-math id="M4">\begin{document}$ \epsilon \to 0 $\end{document}</tex-math></inline-formula> in the nonlinear terms. Therefore, we derive strong two-scale compactness results based on a variational principle. Further, we establish uniqueness for the microscopic and the macroscopic model.</p>