• Locally-periodic media with areas of low and high diffusivity. How can we average them?

Locally-periodic media with areas of low and high diffusivity. How can we average them?

We aim at understanding reaction and transport in those porous materials that present regions with both high and low diffusivities. For such scenarios, the transport becomes structured (here: micro-macro), while the reaction will be mainly hosted by the micro-structures of the low-diffusivity region. The geometry we have in mind include perforations (pores) arranged in a locally-periodic fashion. We choose a prototypical reaction-diffusion system (of minimal size), discuss its formal homogenization ﾐ the heterogenous medium being now assumed to be made of zones with circular areas of low diffusivity of x-varying sizes. We report on two type of results. On one hand, we rely on formal asymptotic homogenization, suitable use of the level sets of the oscillating perforations combined with a boundary unfolding technique to derive the upscaled model equations. On the other hand, we prove the weak solvability of the limit two-scale reaction-diffusion model. A special feature of our analysis is that most of the basic estimates (positivity, boundedness, uniqueness, energy inequality) are obtained in x-dependent Bochner spaces. Finally, the homogenization limit is proven rigorously by means of a suitable corrector estimate (an upper bound on the convergence rate).

This is joint work with Tycho van Noorden (University of Erlangen-Nuremberg, Germany).