MURPHYS-HSFS-2014 - 7th International Workshop on Multi-Rate Processes & Hysteresis, 2nd International Workshop on Hysteresis and Slow-Fast Systems, April 7-11, 2014 - Abstract
Reichelt, Sina
Many reaction-diffusion processes arising in civil engineering, biology, or chemistry take place in strongly heterogeneous media, for instance concrete carbonation or the spread-out of substances in biological tissues. Letting the heterogenities be periodically distributed with microscopic period length $varepsilon > 0$, we are facing difficulties with respect to numerical simulations and the study of pattern formation. It is therefore our aim to rigorously derive effective equations for the limit $varepsilon to 0$. The system under consideration comprises one species with characteristic diffusion length of order $O(1)$ and another with diffusion length $O(varepsilon)$, whereas both species are coupled via nonlinear reaction terms. The slow diffusion of the second species leads to degenerating gradient bounds and hence a lack of compactness, which in turn prevents a straight forward convergence of the nonlinear reaction terms. To overcome the complication of missing compactness and nonlinearities, which was not done before, we employ the method of periodic unfolding and we prove strong two-scale convergence of the slow diffusive species. Finally, we obtain a novel system of coupled reaction-diffusion equations in the two-scale space, which consists of the macroscopic domain and the microscopic unit cell attached to each macroscopic point. Moreover, for smooth given data, we have convergence rates of order $O(varepsilon^1/2)$.