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

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March 2012, 7(1): 1-41. doi: 10.3934/nhm.2012.7.1

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

Xavier Blanc ^1, , Claude Le Bris ^2, and Pierre-Louis Lions ^3,

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, France
3.	Collège de France, 11, place Marcelin Berthelot, 75231 Paris Cedex 05, France

Received September 2011 Revised January 2012 Published February 2012

Abstract
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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.

Keywords: Newton equation, nonlinear wave equation., Macroscopic limit.

Mathematics Subject Classification: Primary: 35Q35, 35Q72, 82C21; Secondary: 35L70, 70F45, 82C2.

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

References:

[1]	Robert A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.
[2]	M. Berezhnyy and L. Berlyand, Continuum limit for three-dimensional mass-spring networks and discrete Korn's inequality, J. Mech. Phys. Solids, 54 (2006), 635-669.
[3]	Susanne C. Brenner, Poincaré-Friedrichs inequalities for piecewise $H^1$ functions, SIAM J. Numer. Anal., 41 (2003), 306-324 (electronic). doi: 10.1137/S0036142902401311.
[4]	Constantine M. Dafermos and William J. Hrusa, Energy methods for quasilinear hyperbolic initial-boundary value problems. Applications to elastodynamics, Arch. Ration. Mech. Anal., 87 (1985), 267-292.
[5]	Wei-nan E and Ping-bing Ming, Cauchy-Born rule and the stability of crystalline solids: Dynamic problems, Acta Math. Appl. Sin. Engl. Ser., 23 (2007), 529-550. doi: 10.1007/s10255-007-0393.
[6]	P. Joly, Finite element methods with continuous displacement, in "Effective Computational Methods for Wave Propagation," Numer. Insights, 5, Chapman & Hall/CRC, Boca Raton, FL, (2008), 267-329.
[7]	Tosio Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Archive for Rational Mechanics and Analysis, 58 (1975), 181-205. doi: 10.1007/BF00280740.
[8]	B. G. Pachpatte, On discrete inequalities of the Poincaré type, Period. Math. Hungar., 19 (1988), 227-233. doi: 10.1007/BF01850291.
[9]	Denis Serre, "Systems of Conservation Laws. 1. Hyperbolicity, Entropies, Shock Waves," Translated from the French by I. N. Sneddon, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9780511612374.
[10]	Antoni Zygmund, "Trigonometrical Series," Dover Publications, New York, 1955.

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References:

[1]	Robert A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.
[2]	M. Berezhnyy and L. Berlyand, Continuum limit for three-dimensional mass-spring networks and discrete Korn's inequality, J. Mech. Phys. Solids, 54 (2006), 635-669.
[3]	Susanne C. Brenner, Poincaré-Friedrichs inequalities for piecewise $H^1$ functions, SIAM J. Numer. Anal., 41 (2003), 306-324 (electronic). doi: 10.1137/S0036142902401311.
[4]	Constantine M. Dafermos and William J. Hrusa, Energy methods for quasilinear hyperbolic initial-boundary value problems. Applications to elastodynamics, Arch. Ration. Mech. Anal., 87 (1985), 267-292.
[5]	Wei-nan E and Ping-bing Ming, Cauchy-Born rule and the stability of crystalline solids: Dynamic problems, Acta Math. Appl. Sin. Engl. Ser., 23 (2007), 529-550. doi: 10.1007/s10255-007-0393.
[6]	P. Joly, Finite element methods with continuous displacement, in "Effective Computational Methods for Wave Propagation," Numer. Insights, 5, Chapman & Hall/CRC, Boca Raton, FL, (2008), 267-329.
[7]	Tosio Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Archive for Rational Mechanics and Analysis, 58 (1975), 181-205. doi: 10.1007/BF00280740.
[8]	B. G. Pachpatte, On discrete inequalities of the Poincaré type, Period. Math. Hungar., 19 (1988), 227-233. doi: 10.1007/BF01850291.
[9]	Denis Serre, "Systems of Conservation Laws. 1. Hyperbolicity, Entropies, Shock Waves," Translated from the French by I. N. Sneddon, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9780511612374.
[10]	Antoni Zygmund, "Trigonometrical Series," Dover Publications, New York, 1955.

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