Homogenization of a locally-periodic medium with areas of low and high diffusivity

Details Homogenization of a locally-periodic medium with areas of low and high diffusivity Homogenization of a locally-periodic medium with areas of low and high diffusivity

2010-03-21

arxiv.org/abs/1003.3979v1

Physics

Mathematical Physics

Scientific paper

We aim at understanding transport in porous materials including regions with both high and low diffusivities. For such scenarios, the transport becomes structured (here: {\em micro-macro}). The geometry we have in mind includes regions of low diffusivity arranged in a locally-periodic fashion. We choose a prototypical advection-diffusion system (of minimal size), discuss its formal homogenization (the heterogenous medium being now assumed to be made of zones with circular areas of low diffusivity of $x$-varying sizes), and 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, $L^\infty$-bounds, uniqueness, energy inequality) are obtained in $x$-dependent Bochner spaces.

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