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

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.
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.

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.

[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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