# American Institute of Mathematical Sciences

December  2013, 6(6): 1487-1506. doi: 10.3934/dcdss.2013.6.1487

## On damping rates of dissipative KdV equations

 1 LAMFA, CNRS UMR 7352, Université de Picardie Jules Verne, Pôle Scienti que, 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

Received  September 2012 Revised  September 2012 Published  April 2013

We consider here different models of dissipative Korteweg-de Vries (KdV) equations on the torus. Using a proper wave function $\Gamma$, we compare numerically the long time behavior effects of the damping models and we propose a hierarchy between these models. We also introduce a method based on the solution of an inverse problem to rebuild a posteriori the damping operator using only samples of the solution.
Citation: Jean-Paul Chehab, Georges Sadaka. On damping rates of dissipative KdV equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1487-1506. doi: 10.3934/dcdss.2013.6.1487
##### 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. doi: 10.1007/s00332-002-0466-4. Google Scholar [2] M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation,, Phys. D, 192 (2004), 265. doi: 10.1016/j.physd.2004.01.023. Google Scholar [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. doi: 10.3934/cpaa.2013.12.519. Google Scholar [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. doi: 10.3934/dcds.2010.27.1473. Google Scholar [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. doi: 10.1016/j.crma.2007.06.007. Google Scholar [6] F. Dubois, Schemes available from:, \url{http://www.math.u-psud.fr/~fdubois/fractionnaire.html} [source Fortran]., (). Google Scholar [7] F. Dubois, A. Galucio and N. Point, "Introduction à la Dérivation Fractionnaire. Théorie et Applications," (in French),, Ref AF510, (2010). Google Scholar [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. doi: 10.1177/1077546307087427. Google Scholar [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, (2012). Google Scholar [10] D. Dutykh, "Modélisation Mathématique des Tsunamis,", (French) [Mathematical modeling of Tsunamis], (2007). Google Scholar [11] D. Dutykh, Visco-potential free-surface flows and long wave modelling,, European Journal of Mechanics B Fluids, 28 (2009), 430. doi: 10.1016/j.euromechflu.2008.11.003. Google Scholar [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. doi: 10.1016/0022-0396(88)90010-1. Google Scholar [13] J.-M. Ghidaglia, A note on the strong convergence towards attractors for damped forced KdV equations,, J. Diff. Eq., 110 (1994), 356. doi: 10.1006/jdeq.1994.1071. Google Scholar [14] O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations,, Discrete Contin. Dynam. Systems, 6 (2000), 625. Google Scholar [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. doi: 10.1006/jdeq.2001.4163. Google Scholar [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. doi: 10.1016/j.cma.2005.10.010. Google Scholar [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, (2012). doi: 10.1080/17415977.2012.720684. Google Scholar [18] C. Hirsch, "Numerical Computation of Internal and External Flows. The Fundamentals of Computational Fluid Dynamics,", Butterworth-Heinemann, (2007). Google Scholar [19] C. Jordan, "Calculus of Finite Differences,", 3rd edition, (1965). Google Scholar [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. doi: 10.1080/03605300903585336. Google Scholar [21] S. K. Lele, Compact finite difference schemes with spectral-like resolution,, J. Comput. Phys., 103 (1992), 16. doi: 10.1016/0021-9991(92)90324-R. Google Scholar [22] A. Miranville and R. Temam, "Mathematical Modeling in Continuum Mechanics,", Second edition, (2005). doi: 10.1017/CBO9780511755422. Google Scholar [23] E. Ott and R. N. Sudan, Nonlinear theory of ion acoustic wave with Landau damping,, Physics of Fluids, 12 (1969), 2388. doi: 10.1063/1.1692358. Google Scholar [24] E. Ott and R. N. Sudan, Damping of solitary waves,, Physics of Fluids, 13 (1970), 1432. doi: 10.1063/1.1693097. Google Scholar [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. doi: 10.3934/dcdsb.2010.14.1511. Google Scholar [26] G. Sadaka, "Etude Mathématique et Numérique d'Équations d'Ondes Aquatiques Amorties,", Thèse de Doctorat, (2011). Google Scholar [27] Lloyd N. Trefethen, "Spectral Methods in MATLAB,", Software, 10 (2000). doi: 10.1137/1.9780898719598. Google Scholar

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. doi: 10.1007/s00332-002-0466-4. Google Scholar [2] M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation,, Phys. D, 192 (2004), 265. doi: 10.1016/j.physd.2004.01.023. Google Scholar [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. doi: 10.3934/cpaa.2013.12.519. Google Scholar [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. doi: 10.3934/dcds.2010.27.1473. Google Scholar [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. doi: 10.1016/j.crma.2007.06.007. Google Scholar [6] F. Dubois, Schemes available from:, \url{http://www.math.u-psud.fr/~fdubois/fractionnaire.html} [source Fortran]., (). Google Scholar [7] F. Dubois, A. Galucio and N. Point, "Introduction à la Dérivation Fractionnaire. Théorie et Applications," (in French),, Ref AF510, (2010). Google Scholar [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. doi: 10.1177/1077546307087427. Google Scholar [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, (2012). Google Scholar [10] D. Dutykh, "Modélisation Mathématique des Tsunamis,", (French) [Mathematical modeling of Tsunamis], (2007). Google Scholar [11] D. Dutykh, Visco-potential free-surface flows and long wave modelling,, European Journal of Mechanics B Fluids, 28 (2009), 430. doi: 10.1016/j.euromechflu.2008.11.003. Google Scholar [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. doi: 10.1016/0022-0396(88)90010-1. Google Scholar [13] J.-M. Ghidaglia, A note on the strong convergence towards attractors for damped forced KdV equations,, J. Diff. Eq., 110 (1994), 356. doi: 10.1006/jdeq.1994.1071. Google Scholar [14] O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations,, Discrete Contin. Dynam. Systems, 6 (2000), 625. Google Scholar [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. doi: 10.1006/jdeq.2001.4163. Google Scholar [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. doi: 10.1016/j.cma.2005.10.010. Google Scholar [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, (2012). doi: 10.1080/17415977.2012.720684. Google Scholar [18] C. Hirsch, "Numerical Computation of Internal and External Flows. The Fundamentals of Computational Fluid Dynamics,", Butterworth-Heinemann, (2007). Google Scholar [19] C. Jordan, "Calculus of Finite Differences,", 3rd edition, (1965). Google Scholar [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. doi: 10.1080/03605300903585336. Google Scholar [21] S. K. Lele, Compact finite difference schemes with spectral-like resolution,, J. Comput. Phys., 103 (1992), 16. doi: 10.1016/0021-9991(92)90324-R. Google Scholar [22] A. Miranville and R. Temam, "Mathematical Modeling in Continuum Mechanics,", Second edition, (2005). doi: 10.1017/CBO9780511755422. Google Scholar [23] E. Ott and R. N. Sudan, Nonlinear theory of ion acoustic wave with Landau damping,, Physics of Fluids, 12 (1969), 2388. doi: 10.1063/1.1692358. Google Scholar [24] E. Ott and R. N. Sudan, Damping of solitary waves,, Physics of Fluids, 13 (1970), 1432. doi: 10.1063/1.1693097. Google Scholar [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. doi: 10.3934/dcdsb.2010.14.1511. Google Scholar [26] G. Sadaka, "Etude Mathématique et Numérique d'Équations d'Ondes Aquatiques Amorties,", Thèse de Doctorat, (2011). Google Scholar [27] Lloyd N. Trefethen, "Spectral Methods in MATLAB,", Software, 10 (2000). doi: 10.1137/1.9780898719598. Google Scholar
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