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On the Cosserat model for thin rods made of thermoelastic materials with voids
On damping rates of dissipative KdV equations
1. | LAMFA, CNRS UMR 7352, Université de Picardie Jules Verne, Pôle Scientique, 33, rue Saint Leu, 80039 Amiens, France |
2. | LAMFA, CNRS UMR 7352, Université de Picardie Jules Verne, Pôle Scientifique, 33, rue Saint Leu, 80039 Amiens, France |
References:
[1] |
J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and the linear theory, J. Nonlinear Sci., 12 (2002), 283-318.
doi: 10.1007/s00332-002-0466-4. |
[2] |
M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation, Phys. D, 192 (2004), 265-278.
doi: 10.1016/j.physd.2004.01.023. |
[3] |
J.-P. Chehab and G. Sadaka, Numerical study of a family of damped KdV equations, Communications on Pure and Applied Analysis,, 12 (2013), 519-546.
doi: 10.3934/cpaa.2013.12.519. |
[4] |
M. Chen, S. Dumont, L. Dupaigne and O. Goubet, Decay of solutions to a water wave model with nonlocal viscous dispersive term, Discrete and Continuous Dynamical Systems, 27 (2010), 1473-1492.
doi: 10.3934/dcds.2010.27.1473. |
[5] |
F. Dias and D. Dutykh, Viscous potentiel free-surface flows in a fluid layer of finite depth, C. R. Math. Acad. Sci. Paris, 345 (2007), 113-118.
doi: 10.1016/j.crma.2007.06.007. |
[6] |
F. Dubois, Schemes available from:, \url{http://www.math.u-psud.fr/~fdubois/fractionnaire.html} [source Fortran]., ().
|
[7] |
F. Dubois, A. Galucio and N. Point, "Introduction à la Dérivation Fractionnaire. Théorie et Applications," (in French), Ref AF510, Techniques de l'ingénieur, April, 2010. |
[8] |
F. Dubois, J.-F. Deü and A. Galucio, The $G^\alpha$-scheme for approximation of fractional derivatives: Application to the dynamics of dissipative systems, J. Vib. Control, 14 (2008), 1597-1605.
doi: 10.1177/1077546307087427. |
[9] |
S. Dumont and J.-B. Duval, Numerical investigation of asymptotical properties of solutions to models for waterwaves with non local viscosity, International Journal of Numerical Analysis and Modeling, to appear, (2012). |
[10] |
D. Dutykh, "Modélisation Mathématique des Tsunamis," (French) [Mathematical modeling of Tsunamis], Ph.D thesis, ENS Cachan, 2007. |
[11] |
D. Dutykh, Visco-potential free-surface flows and long wave modelling, European Journal of Mechanics B Fluids, 28 (2009), 430-443.
doi: 10.1016/j.euromechflu.2008.11.003. |
[12] |
J.-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time, J. Diff. Eq., 74 (1988), 369-390.
doi: 10.1016/0022-0396(88)90010-1. |
[13] |
J.-M. Ghidaglia, A note on the strong convergence towards attractors for damped forced KdV equations, J. Diff. Eq., 110 (1994), 356-359.
doi: 10.1006/jdeq.1994.1071. |
[14] |
O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete Contin. Dynam. Systems, 6 (2000), 625-644. |
[15] |
O. Goubet and R. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, J. Differential Equations, 185 (2002), 25-53.
doi: 10.1006/jdeq.2001.4163. |
[16] |
J. L. Guermond, P. Minev and J. Shen, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Eng., 195 (2006), 6011-6045.
doi: 10.1016/j.cma.2005.10.010. |
[17] |
J. Guerrero, M. Raydan and M. Rojas, A hybrid optimization method for large-scale non-negative full regularization in image restoration, Inverse Problems in Science ad Engineering, to appear, (2012).
doi: 10.1080/17415977.2012.720684. |
[18] |
C. Hirsch, "Numerical Computation of Internal and External Flows. The Fundamentals of Computational Fluid Dynamics," Butterworth-Heinemann, 2007. |
[19] |
C. Jordan, "Calculus of Finite Differences," 3rd edition, Chelsea Publishing Co., New York, 1965. |
[20] |
C. Laurent, L. Rosier and B.-Y. Zhang, Control stabilization of the Korterweg-de Vries equation in a periodic domain, Comm. PDE, 35 (2010), 707-744.
doi: 10.1080/03605300903585336. |
[21] |
S. K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103 (1992), 16-42.
doi: 10.1016/0021-9991(92)90324-R. |
[22] |
A. Miranville and R. Temam, "Mathematical Modeling in Continuum Mechanics," Second edition, Cambridge University Press, Cambridge, 2005.
doi: 10.1017/CBO9780511755422. |
[23] |
E. Ott and R. N. Sudan, Nonlinear theory of ion acoustic wave with Landau damping, Physics of Fluids, 12 (1969), 2388-2394.
doi: 10.1063/1.1692358. |
[24] |
E. Ott and R. N. Sudan, Damping of solitary waves, Physics of Fluids, 13 (1970), 1432-1435.
doi: 10.1063/1.1693097. |
[25] |
A. Pazoto and L. Rosier, Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line, DCDS-B, 14 (2010), 1511-1535.
doi: 10.3934/dcdsb.2010.14.1511. |
[26] |
G. Sadaka, "Etude Mathématique et Numérique d'Équations d'Ondes Aquatiques Amorties," Thèse de Doctorat, Université de Picardie Jules Verne, November, 2011. |
[27] |
Lloyd N. Trefethen, "Spectral Methods in MATLAB," Software, Environments, and Tools, 10, SIAM, Philadelphia, 2000.
doi: 10.1137/1.9780898719598. |
show all references
References:
[1] |
J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and the linear theory, J. Nonlinear Sci., 12 (2002), 283-318.
doi: 10.1007/s00332-002-0466-4. |
[2] |
M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation, Phys. D, 192 (2004), 265-278.
doi: 10.1016/j.physd.2004.01.023. |
[3] |
J.-P. Chehab and G. Sadaka, Numerical study of a family of damped KdV equations, Communications on Pure and Applied Analysis,, 12 (2013), 519-546.
doi: 10.3934/cpaa.2013.12.519. |
[4] |
M. Chen, S. Dumont, L. Dupaigne and O. Goubet, Decay of solutions to a water wave model with nonlocal viscous dispersive term, Discrete and Continuous Dynamical Systems, 27 (2010), 1473-1492.
doi: 10.3934/dcds.2010.27.1473. |
[5] |
F. Dias and D. Dutykh, Viscous potentiel free-surface flows in a fluid layer of finite depth, C. R. Math. Acad. Sci. Paris, 345 (2007), 113-118.
doi: 10.1016/j.crma.2007.06.007. |
[6] |
F. Dubois, Schemes available from:, \url{http://www.math.u-psud.fr/~fdubois/fractionnaire.html} [source Fortran]., ().
|
[7] |
F. Dubois, A. Galucio and N. Point, "Introduction à la Dérivation Fractionnaire. Théorie et Applications," (in French), Ref AF510, Techniques de l'ingénieur, April, 2010. |
[8] |
F. Dubois, J.-F. Deü and A. Galucio, The $G^\alpha$-scheme for approximation of fractional derivatives: Application to the dynamics of dissipative systems, J. Vib. Control, 14 (2008), 1597-1605.
doi: 10.1177/1077546307087427. |
[9] |
S. Dumont and J.-B. Duval, Numerical investigation of asymptotical properties of solutions to models for waterwaves with non local viscosity, International Journal of Numerical Analysis and Modeling, to appear, (2012). |
[10] |
D. Dutykh, "Modélisation Mathématique des Tsunamis," (French) [Mathematical modeling of Tsunamis], Ph.D thesis, ENS Cachan, 2007. |
[11] |
D. Dutykh, Visco-potential free-surface flows and long wave modelling, European Journal of Mechanics B Fluids, 28 (2009), 430-443.
doi: 10.1016/j.euromechflu.2008.11.003. |
[12] |
J.-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time, J. Diff. Eq., 74 (1988), 369-390.
doi: 10.1016/0022-0396(88)90010-1. |
[13] |
J.-M. Ghidaglia, A note on the strong convergence towards attractors for damped forced KdV equations, J. Diff. Eq., 110 (1994), 356-359.
doi: 10.1006/jdeq.1994.1071. |
[14] |
O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete Contin. Dynam. Systems, 6 (2000), 625-644. |
[15] |
O. Goubet and R. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, J. Differential Equations, 185 (2002), 25-53.
doi: 10.1006/jdeq.2001.4163. |
[16] |
J. L. Guermond, P. Minev and J. Shen, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Eng., 195 (2006), 6011-6045.
doi: 10.1016/j.cma.2005.10.010. |
[17] |
J. Guerrero, M. Raydan and M. Rojas, A hybrid optimization method for large-scale non-negative full regularization in image restoration, Inverse Problems in Science ad Engineering, to appear, (2012).
doi: 10.1080/17415977.2012.720684. |
[18] |
C. Hirsch, "Numerical Computation of Internal and External Flows. The Fundamentals of Computational Fluid Dynamics," Butterworth-Heinemann, 2007. |
[19] |
C. Jordan, "Calculus of Finite Differences," 3rd edition, Chelsea Publishing Co., New York, 1965. |
[20] |
C. Laurent, L. Rosier and B.-Y. Zhang, Control stabilization of the Korterweg-de Vries equation in a periodic domain, Comm. PDE, 35 (2010), 707-744.
doi: 10.1080/03605300903585336. |
[21] |
S. K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103 (1992), 16-42.
doi: 10.1016/0021-9991(92)90324-R. |
[22] |
A. Miranville and R. Temam, "Mathematical Modeling in Continuum Mechanics," Second edition, Cambridge University Press, Cambridge, 2005.
doi: 10.1017/CBO9780511755422. |
[23] |
E. Ott and R. N. Sudan, Nonlinear theory of ion acoustic wave with Landau damping, Physics of Fluids, 12 (1969), 2388-2394.
doi: 10.1063/1.1692358. |
[24] |
E. Ott and R. N. Sudan, Damping of solitary waves, Physics of Fluids, 13 (1970), 1432-1435.
doi: 10.1063/1.1693097. |
[25] |
A. Pazoto and L. Rosier, Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line, DCDS-B, 14 (2010), 1511-1535.
doi: 10.3934/dcdsb.2010.14.1511. |
[26] |
G. Sadaka, "Etude Mathématique et Numérique d'Équations d'Ondes Aquatiques Amorties," Thèse de Doctorat, Université de Picardie Jules Verne, November, 2011. |
[27] |
Lloyd N. Trefethen, "Spectral Methods in MATLAB," Software, Environments, and Tools, 10, SIAM, Philadelphia, 2000.
doi: 10.1137/1.9780898719598. |
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