Two-scale homogenization of nonlinear
   reaction-diffusion systems with slow diffusion

Alexander Mielke; Sina Reichelt; Marita Thomas; Alexander Mielke; Sina Reichelt; Marita Thomas

doi:10.3934/nhm.2014.9.353

Networks and Heterogeneous Media

2014, Volume 9, Issue 2: 353-382. doi: 10.3934/nhm.2014.9.353

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Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion

1.
Weierstrass Institute, Mohrenstraße 39, 10117 Berlin
2.
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin

Received: 01 November 2013 Revised: 01 April 2014
35B25, 35A01, 35K57, 35K65, 35M10.

We derive a two-scale homogenization limit for reaction-diffusion systems where for some species the diffusion length is of order 1 whereas for the other species the diffusion length is of the order of the periodic microstructure. Thus, in the limit the latter species will display diffusion only on the microscale but not on the macroscale. Because of this missing compactness, the nonlinear coupling through the reaction terms cannot be homogenized but needs to be treated on the two-scale level. In particular, we have to develop new error estimates to derive strong convergence results for passing to the limit.
- Two-scale convergence,
- folding and unfolding,
- coupled reaction-diffusion equations,
- nonlinear reaction,
- degenerating diffusion,
- Gronwall estimate
Citation: Alexander Mielke, Sina Reichelt, Marita Thomas. Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion[J]. Networks and Heterogeneous Media, 2014, 9(2): 353-382. doi: 10.3934/nhm.2014.9.353

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Abstract

We derive a two-scale homogenization limit for reaction-diffusion systems where for some species the diffusion length is of order 1 whereas for the other species the diffusion length is of the order of the periodic microstructure. Thus, in the limit the latter species will display diffusion only on the microscale but not on the macroscale. Because of this missing compactness, the nonlinear coupling through the reaction terms cannot be homogenized but needs to be treated on the two-scale level. In particular, we have to develop new error estimates to derive strong convergence results for passing to the limit.

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