The concept of two-scale convergence associated with a fixed periodic Borel measure is introduced. In the case when is Lebesgue measure on the torus convergence in the sense of Nguetseng-Allaire is obtained. The main properties of two-scale convergence are revealed by the simultaneous consideration of a sequence of functions and a sequence of their gradients. An application of two-scale convergence to the homogenization of some problems in the theory of porous media (the double-porosity model) is presented. A mathematical notion of softly or weakly coupled parallel flows is worked out. A homogenized operator is constructed and the convergence result itself is interpreted as a strong two-scale resolvent convergence. Problems concerning the behaviour of the spectrum under homogenization are touched upon in this connection.

On an extension of the method of two-scale convergence and its applications

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Об одном расширении и применении метода двухмасштабной сходимости@@@On an extension of the method of two-scale convergence and its applications

On considere la diffusion, la convection et les reactions chimiques non lineaires a travers un arrangement periodique de cellules. On suppose que chaque cellule contient des corps poreux entoures de membranes semi-permeables, c'est-a-dire qu'il y a liaison non lineaire entre les flux et les concentrations sur les interfaces cellules-fluide. Nous considerons la limite, quand le nombre de cellules tend vers l'infini et en meme temps lorsque leur volume tend vers zero, tout en conservant la meme fraction de volume. Notre probleme initial se comporte a grande echelle comme un probleme non lineaire ou interviennent deux echelles d'espace, ― une echelle globale et une echelle locale

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Reactive transport through an array of cells with semi-permeable membranes

Diffusion models for fractured media

We study the problem of diffusive transport of biomolecules in the intercellular space, modeled as porous medium, and of their binding to the receptors located on the surface membranes of the cells. Cells are distributed periodically in a bounded domain. To describe this process we introduce a reaction-diffusion equation coupled with nonlinear ordinary differential equations on the boundary. We prove existence and uniqueness of the solution of this problem. We consider the limit, when the number of cells tends to infinity and at the same time their size tends to zero, while the volume fraction of the cells remains fixed. Using the homogenization technique of two-scale convergence, we show that the sequence of solutions of the original problem converges to the solution of the so-called macroscopic problem. To show the convergence of the nonlinear terms on the surfaces we use the unfolding method (periodic modulation). We discuss applicability of the result to mathematical description of membrane receptors ...

Ulrich Hornung

Papers

Diffusion, convection, adsorption, and reaction of chemicals in porous media