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two-scale model

European Journal of Applied Mathematics (1)

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Homogenisation of a locally periodic medium with areas of low and high diffusivity
T. L. VAN NOORDEN, A. MUNTEAN
Journal:

European Journal of Applied Mathematics / Volume 22 / Issue 5 / October 2011

Published online by Cambridge University Press:

17 May 2011, pp. 493-516
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We aim at understanding transport in porous materials consisting of regions with both high and low diffusivities. We apply a formal homogenisation procedure to the case where the heterogeneities are not arranged in a strictly periodic manner. The result is a two-scale model formulated in x-dependent Bochner spaces. We prove the weak solvability of the limit two-scale model for a prototypical advection–diffusion system of minimal size. A special feature of our analysis is that most of the basic estimates (positivity, L∞-bounds, uniqueness, energy inequality) are obtained in the x-dependent Bochner spaces.