-
Previous Article
Differential equation approximations of stochastic network processes: An operator semigroup approach
- NHM Home
- This Issue
- Next Article
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 |
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. |
[1] |
Kamel Hamdache, Djamila Hamroun. Macroscopic limit of the kinetic Bloch equation. Kinetic and Related Models, 2021, 14 (3) : 541-570. doi: 10.3934/krm.2021015 |
[2] |
Tai-Chia Lin. Vortices for the nonlinear wave equation. Discrete and Continuous Dynamical Systems, 1999, 5 (2) : 391-398. doi: 10.3934/dcds.1999.5.391 |
[3] |
Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations and Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013 |
[4] |
Anatoli Babin, Alexander Figotin. Newton's law for a trajectory of concentration of solutions to nonlinear Schrodinger equation. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1685-1718. doi: 10.3934/cpaa.2014.13.1685 |
[5] |
Q-Heung Choi, Tacksun Jung. A nonlinear wave equation with jumping nonlinearity. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 797-802. doi: 10.3934/dcds.2000.6.797 |
[6] |
Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099 |
[7] |
Jorge A. Esquivel-Avila. Qualitative analysis of a nonlinear wave equation. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 787-804. doi: 10.3934/dcds.2004.10.787 |
[8] |
Yan Yong, Weiyuan Zou. Macroscopic regularity for the relativistic Boltzmann equation with initial singularities. Kinetic and Related Models, 2019, 12 (5) : 945-967. doi: 10.3934/krm.2019036 |
[9] |
Sebastian Engel, Karl Kunisch. Optimal control of the linear wave equation by time-depending BV-controls: A semi-smooth Newton approach. Mathematical Control and Related Fields, 2020, 10 (3) : 591-622. doi: 10.3934/mcrf.2020012 |
[10] |
H. A. Erbay, S. Erbay, A. Erkip. The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6101-6116. doi: 10.3934/dcds.2016066 |
[11] |
Aníbal Rodríguez-Bernal, Enrique Zuazua. Parabolic singular limit of a wave equation with localized boundary damping. Discrete and Continuous Dynamical Systems, 1995, 1 (3) : 303-346. doi: 10.3934/dcds.1995.1.303 |
[12] |
Aiyong Chen, Chi Zhang, Wentao Huang. Limit speed of traveling wave solutions for the perturbed generalized KdV equation. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022048 |
[13] |
Liu Rui. The explicit nonlinear wave solutions of the generalized $b$-equation. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1029-1047. doi: 10.3934/cpaa.2013.12.1029 |
[14] |
Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems and Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121 |
[15] |
Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure and Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165 |
[16] |
Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control and Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017 |
[17] |
Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223 |
[18] |
Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159 |
[19] |
Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control and Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73 |
[20] |
Kim Dang Phung. Decay of solutions of the wave equation with localized nonlinear damping and trapped rays. Mathematical Control and Related Fields, 2011, 1 (2) : 251-265. doi: 10.3934/mcrf.2011.1.251 |
2020 Impact Factor: 1.213
Tools
Metrics
Other articles
by authors
[Back to Top]