Homogenization of first order equations with $u/\epsilon$-periodic Hamiltonians. Part II: application to dislocations dynamics

Cyril Imbert (1) , Régis Monneau (2) , Elisabeth Rouy (3)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique
3 ECL - École Centrale de Lyon

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.

Mots clés: periodic homogenization; Hamilton-Jacobi equations; integro-differential operators; dislocations dynamics; non-periodic approximate correctors

Domaines: Equations aux dérivées partielles [math.AP]

Cyril Imbert, Régis Monneau, Elisabeth Rouy. Homogenization of first order equations with $u/\epsilon$-periodic Hamiltonians. Part II: application to dislocations dynamics. Communications in Partial Differential Equations, 2008, 33 (3), pp.479-516. ⟨10.1080/03605300701318922⟩. ⟨hal-00080397v2⟩