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We aim at understanding transport in porous materials including regions withboth high and low diffusivities. For such scenarios, the transport becomesstructured (here: {\em micro-macro}). The geometry we have in mind includesregions of low diffusivity arranged in a locally-periodic fashion. We choose aprototypical advection-diffusion system (of minimal size), discuss its formalhomogenization (the heterogenous medium being now assumed to be made of zoneswith circular areas of low diffusivity of $x$-varying sizes), and prove theweak solvability of the limit two-scale reaction-diffusion model. A specialfeature of our analysis is that most of the basic estimates (positivity,$L^\infty$-bounds, uniqueness, energy inequality) are obtained in $x$-dependentBochner spaces.

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