Homogenization of fully overdamped Frenkel-Kontorova models

Monneau, Régis; Imbert, Cyril; Forcadel, Nicolas

Monneau, Régis; Imbert, Cyril; Forcadel, Nicolas (2009), Homogenization of fully overdamped Frenkel-Kontorova models, Journal of Differential Equations, 246, p. 1057-1097. http://dx.doi.org/10.1016/j.jde.2008.06.034

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00266994/en/

Date

2009

Journal name

Journal of Differential Equations

Volume

246

Publisher

Elsevier

Pages

1057-1097

Abstract (EN)

In this paper, we consider the fully overdamped Frenkel-Kontorova model. This is an infinite system of coupled first order ODEs. Each ODE represents the microscopic evolution of one particle interacting with its neighbors and submitted to a fixed periodic potential. After a proper rescaling, a macroscopic model describing the evolution of densities of particles is obtained. We get this homogenization result for a general class of Frenkel-Kontorova models. The proof is based on the construction of suitable hull functions in the framework of viscosity solutions.

Subjects / Keywords

Slepcev formulation; cumulative distribution function; hull function; Hamilton-Jacobi equations; Frenkel-Kontorova models; periodic homogenization; Particle systems