Homogenization of Reaction--Diffusion Processes in a Two-Component Porous Medium with Nonlinear Flux Conditions at the Interface

Abstract

In this paper, we are dealing with the mathematical modeling and homogenization of nonlinear reaction-diffusion processes in a porous medium that consists of two components separated by an interface. One of the components is connected, and the other one is disconnected and consists of periodically distributed inclusions. At the interface, the fluxes are given by nonlinear functions of the concentrations on both sides of the interface. Thus, the concentrations may be discontinuous across the interface. For the derivation of the effective (homogenized) model, we use the method of two-scale convergence. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we give a new approach which involves the boundary unfolding operator and a compactness result for Banach-space-valued functions. The model is motivated by metabolic and regulatory processes in cells, where biochemical species are exchanged between organelles and cytoplasm through the organellar membranes. In this context the nonlinearities are given by kinetics corresponding to multispecies enzyme catalyzed reactions, which are generalizations of the classical Michaelis--Menten kinetics to multispecies reactions.