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

Xavier Blanc; Claude Le Bris; Pierre-Louis Lions; Xavier Blanc; Claude Le Bris; Pierre-Louis Lions

doi:10.3934/nhm.2012.7.1

Networks and Heterogeneous Media

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: 01 September 2011 Revised: 01 January 2012
Primary: 35Q35, 35Q72, 82C21; Secondary: 35L70, 70F45, 82C22.

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.
- Macroscopic limit,
- Newton equation,
- nonlinear wave equation
Citation: Xavier Blanc, Claude Le Bris, Pierre-Louis Lions. From the Newton equation to the wave equation in some simple cases[J]. Networks and Heterogeneous Media, 2012, 7(1): 1-41. doi: 10.3934/nhm.2012.7.1

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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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