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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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B. G. Pachpatte, On discrete inequalities of the Poincaré type, Period. Math. Hungar., 19 (1988), 227-233. doi: 10.1007/BF01850291.  Google Scholar

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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.  Google Scholar

[10]

Antoni Zygmund, "Trigonometrical Series," Dover Publications, New York, 1955.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

B. G. Pachpatte, On discrete inequalities of the Poincaré type, Period. Math. Hungar., 19 (1988), 227-233. doi: 10.1007/BF01850291.  Google Scholar

[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.  Google Scholar

[10]

Antoni Zygmund, "Trigonometrical Series," Dover Publications, New York, 1955.  Google Scholar

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