Homogenization of the dislocation dynamics and of some particle systems with two-body interactions

Nicolas Forcadel (1) , Cyril Imbert (2) , Régis Monneau (1)

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

This paper is concerned with the homogenization of a non-local first order Hamilton-Jacobi equation describing the dynamics of several dislocation lines and the homogenization of some particle systems with two-body interactions. The first objective is to establish a connection between the rescaled dynamics of a increasing number of dislocation lines and the dislocation dynamics density, passing from a discrete model (dislocation lines) to a continuous one (dislocation density). A first answer to this problem was presented in a paper by Rouy and the two last authors \cite{imr2} but the geometric definition of the fronts was not completely satisfactory. This problem is completely solved here. The limit equation is a nonlinear diffusion equation involving a first order Lévy operator. This integral operator keeps memory of the long range interactions, while the nonlinearity keeps memory of short ones. The techniques and tools we introduce turn out to be the right ones to get homogenization results for the dynamics of particles in two-body interaction. The systems of ODEs we consider are very close to overdamped Frenkel-Kontorova models. We prove that the rescaled ``cumulative distribution function'' of the particles converges towards the continuous solution of a nonlinear diffusion equation.

Mots clés: Frenkel-Kontorova model; periodic homogenization; Hamilton-Jacobi equations; moving fronts; two-body interactions; integro-differential operators; Lévy operator; dislocation dynamics; Slepcev formulation; Frenkel-Kontorova model.

Domaines: Equations aux dérivées partielles [math.AP]

Nicolas Forcadel, Cyril Imbert, Régis Monneau. Homogenization of the dislocation dynamics and of some particle systems with two-body interactions. 2007. ⟨hal-00140545v2⟩