From the Newton equation to the wave equation in some simple cases

Xavier Blanc; Claude Le Bris; Pierre-Louis Lions

doi:10.3934/nhm.2012.7.1

Article Contents

2012, Volume 7, Issue 1: 1-41. Doi: 10.3934/nhm.2012.7.1

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From the Newton equation to the wave equation in some simple cases

1.
CEA, DAM, DIF, F-91297, Arpajon
2.
École Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2
3.
Collège de France, 11, place Marcelin Berthelot, 75231 Paris Cedex 05

Received: August 31, 2011

Revised: December 31, 2011

Published: February 2012

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Abstract

We prove that, in some simple situations at least, the one-dimensional wave equation is the limit as the microscopic scale goes to zero of some time-dependent Newton type equation of motion for atomistic systems. We address both some linear and some nonlinear cases.

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Mathematics Subject Classification: Primary: 35Q35, 35Q72, 82C21; Secondary: 35L70, 70F45, 82C22.

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