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

Abstract : 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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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, Taylor & Francis, 2008, 33 (3), pp.479-516. ⟨10.1080/03605300701318922⟩. ⟨hal-00080397v2⟩

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