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

dc.contributor.authorForcadel, Nicolas
HAL ID: 171794
ORCID: 0000-0003-4141-8385
dc.contributor.authorImbert, Cyril
HAL ID: 9368
ORCID: 0000-0002-1290-8257
dc.contributor.authorMonneau, Régis
dc.date.accessioned2009-07-01T12:46:50Z
dc.date.available2009-07-01T12:46:50Z
dc.date.issued2009
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/609
dc.language.isoenen
dc.subjectLévy operatoren
dc.subjectperiodic homogenizationen
dc.subjectHamilton-Jacobi equationsen
dc.subjectmoving frontsen
dc.subjecttwo-body interactionsen
dc.subjectintegro-differential operatorsen
dc.subjectdislocation dynamicsen
dc.subjectSlepcev formulationen
dc.subjectFrenkel-Kontorova modelen
dc.subjectparticle systemsen
dc.subject.ddc515en
dc.titleHomogenization of some particle systems with two-body interactions and of the dislocation dynamicsen
dc.typeArticle accepté pour publication ou publié
dc.contributor.editoruniversityotherINRIA – Ecole Nationale des Ponts et Chaussées;France
dc.description.abstractenThis paper is concerned with the homogenization of some particle systems with two-body interactions in dimension one and of dislocation dynamics in higher dimensions. The dynamics of our particle systems are described by some ODEs. We prove that the rescaled ``cumulative distribution function'' of the particles converges towards the solution of a Hamilton-Jacobi equation. In the case when the interactions between particles have a slow decay at infinity as $1/x$, we show that this Hamilton-Jacobi equation contains an extra diffusion term which is a half Laplacian. We get the same result in the particular case where the repulsive interactions are exactly $1/x$, which creates some additional difficulties at short distances. We also study a higher dimensional generalisation of these particle systems which is particularly meaningful to describe the dynamics of dislocations lines. One main result of this paper is the discovery of a satisfactory mathematical formulation of this dynamics, namely a Slepcev formulation. We show in particular that the system of ODEs for particle systems can be naturally imbedded in this Slepcev formulation. Finally, with this formulation in hand, we get homogenization results which contain the particular case of particle systems.en
dc.relation.isversionofjnlnameDiscrete and Continuous Dynamical Systems. Series A
dc.relation.isversionofjnlvol23en
dc.relation.isversionofjnlissue3en
dc.relation.isversionofjnldate2009-03
dc.relation.isversionofjnlpages785-826en
dc.relation.isversionofdoihttp://dx.doi.org/10.3934/dcds.2009.23.785en
dc.identifier.urlsitehttp://hal.archives-ouvertes.fr/hal-00140545/en/en
dc.description.sponsorshipprivateouien
dc.relation.isversionofjnlpublisherAmerican Institute of Mathematical Sciencesen
dc.subject.ddclabelAnalyseen


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