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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Homogenization of accelerated Frenkel-Kontorova models with $ n$ types of particles


Authors: N. Forcadel, C. Imbert and R. Monneau
Journal: Trans. Amer. Math. Soc. 364 (2012), 6187-6227
MSC (2010): Primary 35B27, 35F20, 45K05, 47G20, 49L25, 35B10
Published electronically: July 10, 2012
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Abstract: We consider systems of ODEs that describe the dynamics of particles. Each particle satisfies a Newton law (including a damping term and an acceleration term) where the force is created by the interactions with other particles and with a periodic potential. The presence of a damping term allows the system to be monotone. Our study takes into account the fact that the particles can be different.

After a proper hyperbolic rescaling, we show that solutions of these systems of ODEs converge to solutions of some macroscopic homogenized Hamilton-Jacobi equations.


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Additional Information

N. Forcadel
Affiliation: CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cedex 16, France

C. Imbert
Affiliation: CNRS, UMR 8050, Centre de Mathématiques, Université Paris-Est Créteil, Val de Marne P3, 61 Avenue du Général de Gaulle, 94010 Créteil Cedex France

R. Monneau
Affiliation: Cermics, Universite Paris-Est, Ecole des ponts, 6-8 avenue Blaise Pascal, 77455 Marne la Vallee Cedex 2, France

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05650-9
PII: S 0002-9947(2012)05650-9
Keywords: Particle system, periodic homogenization, Frenkel-Kontorova models, Hamilton-Jacobi equations, hull function
Received by editor(s): June 30, 2009
Received by editor(s) in revised form: June 6, 2010
Published electronically: July 10, 2012
Article copyright: © Copyright 2012 American Mathematical Society