no. 6
Partial differential equations
A convergence result for the periodic unfolding method related to fast diffusion on manifolds
[Un résultat de convergence pour la méthode d'éclatement périodique lié à la diffusion rapide sur des variétés]

Présenté par : Alain Bensoussan

Graf, Isabell1 ; Peter, Malte A.2, 3
1 Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada
2 Institute of Mathematics, University of Augsburg, Augsburg, Germany
3 Augsburg Centre for Innovative Technologies, Augsburg, Germany
Comptes Rendus. Mathématique, Tome 352 (2014) no. 6, pp. 485-490.

À l'aide de la méthode d'éclatement périodique, nous démontrons un résultat de convergence des gradients de fonctions définies sur des variétés connexes, différentiables et périodiques. Sous certaines conditions d'estimation du gradient, typiques de la diffusion rapide, nous obtenons à la limite d'homogénéisation la somme d'un gradient de la variable globale et d'un gradient de la variable locale. Un exemple illustre l'utilisation de ce résultat : pour une équation de réaction et diffusion définie sur une variété périodique, nous démontrons que l'équation homogénéisée contient un terme décrivant une diffusion globale.

Based on the periodic unfolding method in periodic homogenization, we deduce a convergence result for gradients of functions defined on connected, smooth, and periodic manifolds. Under the assumption of certain a-priori estimates of the gradient, which are typical for fast diffusion, the sum of a term involving a gradient with respect to the slow variable and one with respect to the fast variable is obtained in the homogenization limit. In addition, we show in a brief example how to apply this result and find for a reaction–diffusion equation defined on a periodic manifold that the homogenized equation contains a term describing macroscopic diffusion.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.03.002
Graf, Isabell 1 ; Peter, Malte A. 2, 3

1 Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada
2 Institute of Mathematics, University of Augsburg, Augsburg, Germany
3 Augsburg Centre for Innovative Technologies, Augsburg, Germany 
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Graf, Isabell; Peter, Malte A. A convergence result for the periodic unfolding method related to fast diffusion on manifolds. Comptes Rendus. Mathématique, Tome 352 (2014) no. 6, pp. 485-490. doi : 10.1016/j.crma.2014.03.002. http://www.numdam.org/articles/10.1016/j.crma.2014.03.002/

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