Homogenization of First-Order Equations with (u/ε) -Periodic Hamiltonians. Part II: application to dislocations dynamics
Imbert, Cyril; Monneau, Régis; Rouy, Elisabeth (2007), Homogenization of First-Order Equations with (u/ε) -Periodic Hamiltonians. Part II: application to dislocations dynamics. https://basepub.dauphine.fr/handle/123456789/12970
TypeDocument de travail / Working paper
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Abstract (EN)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=". 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 di usion 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.
Subjects / Keywordsperiodic homogenization; integro-differential operators; Hamilton-Jacobi equations; dislocations dynamics; non-periodic approximate correctors
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