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| HAL : hal-00080397, version 2 |
| Fiche détaillée |  Récupérer au format |
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| Versions disponibles : | v1 (28-06-2006) | v2 (16-02-2007) |
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| Homogenization of first order equations with $u/\epsilon$-periodic Hamiltonians. Part II: application to dislocations dynamics |
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| Cyril Imbert 1Régis Monneau 2 |
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| ACI JC 1025 Collaboration(s) |
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| (16/06/2006) |
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| This paper is concerned with a result of homogenization of a non-local first order Hamilton-Jacobi equations describing the dislocations dynamics. Our model for the interaction between dislocations involve both an integro-differential operator and a (local) Hamiltonian depending periodicly on $u/\eps$. The first two authors studied in a previous work homogenization problems involving such local Hamiltonians. Two main ideas of this previous work are used: on the one hand, we prove an ergodicity property of this equation by constructing approximate correctors which are necessarily non periodic in space in general; on the other hand, the proof of the convergence of the solution uses here a twisted perturbed test function for a higher dimensional problem. The limit equation is a nonlinear diffusion equation involving a first order Lévy operator; the nonlinearity keeps memory of the short range interaction, while the Lévy operator keeps memory of long ones. The homogenized equation is a kind of effective plastic law for densities of dislocations moving in a single slip plane. |
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| 1 : | Institut de Mathématiques et de Modélisation de Montpellier (I3M) |
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CNRS : UMR5149 – Université Montpellier II - Sciences et Techniques du Languedoc |
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| 2 : | Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique (CERMICS) |
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INRIA – Ecole des Ponts ParisTech |
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| 3 : | École Centrale de Lyon (ECL) |
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Ministère de l'Enseignement Supérieur et de la Recherche Scientifique |
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| EDP et Matériaux |
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| Domaine | : | Mathématiques/Equations aux dérivées partielles |
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| periodic homogenization – Hamilton-Jacobi equations – integro-differential operators – dislocations dynamics – non-periodic approximate correctors |
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| Liste des fichiers attachés à ce document : | |||||
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| hal-00080397, version 2 | |
| http://hal.archives-ouvertes.fr/hal-00080397 | |
| oai:hal.archives-ouvertes.fr:hal-00080397 | |
| Contributeur : Cyril Imbert | |
| Soumis le : Vendredi 16 Février 2007, 14:13:11 | |
| Dernière modification le : Vendredi 16 Février 2007, 15:04:06 | |
