November  2013, 12(6): 2669-2684. doi: 10.3934/cpaa.2013.12.2669

Long time dynamics for forced and weakly damped KdV on the torus

1. 

Department of Mathematics, University of Illinois, Urbana, IL 61801, United States

2. 

Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL, 61801

Received  August 2012 Revised  April 2013 Published  May 2013

The forced and weakly damped Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. Starting from $L^2$ and mean-zero initial data we prove that the solution decomposes into two parts; a linear one which decays to zero as time goes to infinity and a nonlinear one which always belongs to a smoother space. As a corollary we prove that all solutions are attracted by a ball in $H^s$, $s\in(0,1)$, whose radius depends only on $s$, the $L^2$ norm of the forcing term and the damping parameter. This gives a new proof for the existence of a smooth global attractor and provides quantitative information on the size of the attractor set in $H^s$. In addition we prove that higher order Sobolev norms are bounded for all positive times.
Citation: M. Burak Erdoğan, Nikolaos Tzirakis. Long time dynamics for forced and weakly damped KdV on the torus. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2669-2684. doi: 10.3934/cpaa.2013.12.2669
References:
[1]

A. V. Babin, A. A. Ilyin and E. S. Titi, On the regularization mechanism for the periodic Korteweg-de Vries equation,, Comm. Pure Appl. Math., 64 (2011), 591.  doi: 10.1002/cpa.20356.  Google Scholar

[2]

J. M. Ball, Global attractors for damped semilinear wave equations,, Discrete Contin. Dyn. Syst., 10 (2003), 31.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[3]

J. Bourgain, Fourier transform restriction phenomena for cer tain lattice subsets and applications to nonlinear evolution equations. Part II: The KdV equation,, GAFA, 3 (1993), 209.  doi: 10.1007/BF01895688.  Google Scholar

[4]

M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation,, Physica D, 192 (2004), 265.  doi: 10.1016/j.physd.2004.01.023.  Google Scholar

[5]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp Global Well-Posedness for KdV and Modified KdV on $\mathbb R$ and $\mathbb T$,, J. Amer. Math. Soc., 16 (2003), 705.  doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

[6]

M. B. Erdoğan and N. Tzirakis, Global smoothing for the periodic KdV evolution,, to appear in Inter. Math. Res. Not., ().   Google Scholar

[7]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system,, J. Functional Analysis, 151 (1997), 384.  doi: 10.1006/jfan.1997.3148.  Google Scholar

[8]

J. M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time,, J. Diff. Eqs., 74 (1988), 369.  doi: 10.1016/0022-0396(88)90010-1.  Google Scholar

[9]

J. M. Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations,, J. Diff. Eqs., 110 (1994), 356.  doi: 10.1006/jdeq.1994.1071.  Google Scholar

[10]

O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations,, Discrete Contin. Dyn. Syst., 6 (2000), 625.  doi: 10.1006/jdeq.2000.3763.  Google Scholar

[11]

T. Kappeler and P. Topalov, Global wellposedness of KdV in $H^{-1}(T, R)$,, Duke Math. J., 135 (2006), 327.  doi: 10.1215/S0012-7094-06-13524-X.  Google Scholar

[12]

C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation,, J. Amer. Math. Soc., 9 (1996), 573.  doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar

[13]

S. B. Kuksin, "Analysis of Hamiltonian PDEs,", Oxford University Press, (2000).   Google Scholar

[14]

J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations,, Comm. Pure Appl. Math., 38 (1985), 685.  doi: 10.1002/cpa.3160380516.  Google Scholar

[15]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Phyiscs,", Applied Mathematical Sciences, 68 (1997).   Google Scholar

[16]

K. Tsugawa, Existence of the global attractor for weakly damped forced KdV equation on Sobolev spaces of negative index,, Commun. Pure Appl. Anal., 3 (2004), 301.  doi: 10.3934/cpaa.2004.3.301.  Google Scholar

[17]

X. Yang, Global attractor for the weakly damped forced KdV equation in Sobolev spaces of low regularity,, Nonlinear Differ. Equ. Appl., 18 (2011), 273.  doi: 10.1007/s00030-010-0095-9.  Google Scholar

show all references

References:
[1]

A. V. Babin, A. A. Ilyin and E. S. Titi, On the regularization mechanism for the periodic Korteweg-de Vries equation,, Comm. Pure Appl. Math., 64 (2011), 591.  doi: 10.1002/cpa.20356.  Google Scholar

[2]

J. M. Ball, Global attractors for damped semilinear wave equations,, Discrete Contin. Dyn. Syst., 10 (2003), 31.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[3]

J. Bourgain, Fourier transform restriction phenomena for cer tain lattice subsets and applications to nonlinear evolution equations. Part II: The KdV equation,, GAFA, 3 (1993), 209.  doi: 10.1007/BF01895688.  Google Scholar

[4]

M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation,, Physica D, 192 (2004), 265.  doi: 10.1016/j.physd.2004.01.023.  Google Scholar

[5]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp Global Well-Posedness for KdV and Modified KdV on $\mathbb R$ and $\mathbb T$,, J. Amer. Math. Soc., 16 (2003), 705.  doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

[6]

M. B. Erdoğan and N. Tzirakis, Global smoothing for the periodic KdV evolution,, to appear in Inter. Math. Res. Not., ().   Google Scholar

[7]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system,, J. Functional Analysis, 151 (1997), 384.  doi: 10.1006/jfan.1997.3148.  Google Scholar

[8]

J. M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time,, J. Diff. Eqs., 74 (1988), 369.  doi: 10.1016/0022-0396(88)90010-1.  Google Scholar

[9]

J. M. Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations,, J. Diff. Eqs., 110 (1994), 356.  doi: 10.1006/jdeq.1994.1071.  Google Scholar

[10]

O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations,, Discrete Contin. Dyn. Syst., 6 (2000), 625.  doi: 10.1006/jdeq.2000.3763.  Google Scholar

[11]

T. Kappeler and P. Topalov, Global wellposedness of KdV in $H^{-1}(T, R)$,, Duke Math. J., 135 (2006), 327.  doi: 10.1215/S0012-7094-06-13524-X.  Google Scholar

[12]

C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation,, J. Amer. Math. Soc., 9 (1996), 573.  doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar

[13]

S. B. Kuksin, "Analysis of Hamiltonian PDEs,", Oxford University Press, (2000).   Google Scholar

[14]

J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations,, Comm. Pure Appl. Math., 38 (1985), 685.  doi: 10.1002/cpa.3160380516.  Google Scholar

[15]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Phyiscs,", Applied Mathematical Sciences, 68 (1997).   Google Scholar

[16]

K. Tsugawa, Existence of the global attractor for weakly damped forced KdV equation on Sobolev spaces of negative index,, Commun. Pure Appl. Anal., 3 (2004), 301.  doi: 10.3934/cpaa.2004.3.301.  Google Scholar

[17]

X. Yang, Global attractor for the weakly damped forced KdV equation in Sobolev spaces of low regularity,, Nonlinear Differ. Equ. Appl., 18 (2011), 273.  doi: 10.1007/s00030-010-0095-9.  Google Scholar

[1]

Olivier Goubet. Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 625-644. doi: 10.3934/dcds.2000.6.625

[2]

Eduardo Cerpa. Control of a Korteweg-de Vries equation: A tutorial. Mathematical Control & Related Fields, 2014, 4 (1) : 45-99. doi: 10.3934/mcrf.2014.4.45

[3]

M. Agrotis, S. Lafortune, P.G. Kevrekidis. On a discrete version of the Korteweg-De Vries equation. Conference Publications, 2005, 2005 (Special) : 22-29. doi: 10.3934/proc.2005.2005.22

[4]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain. Mathematical Control & Related Fields, 2011, 1 (3) : 353-389. doi: 10.3934/mcrf.2011.1.353

[5]

Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255

[6]

Muhammad Usman, Bing-Yu Zhang. Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1509-1523. doi: 10.3934/dcds.2010.26.1509

[7]

Eduardo Cerpa, Emmanuelle Crépeau. Rapid exponential stabilization for a linear Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 655-668. doi: 10.3934/dcdsb.2009.11.655

[8]

Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069

[9]

Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61

[10]

Netra Khanal, Ramjee Sharma, Jiahong Wu, Juan-Ming Yuan. A dual-Petrov-Galerkin method for extended fifth-order Korteweg-de Vries type equations. Conference Publications, 2009, 2009 (Special) : 442-450. doi: 10.3934/proc.2009.2009.442

[11]

Ludovick Gagnon. Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1489-1507. doi: 10.3934/dcds.2017061

[12]

Arnaud Debussche, Jacques Printems. Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 761-781. doi: 10.3934/dcdsb.2006.6.761

[13]

Qifan Li. Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1097-1109. doi: 10.3934/cpaa.2012.11.1097

[14]

Brian Pigott. Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 389-418. doi: 10.3934/cpaa.2014.13.389

[15]

Anne de Bouard, Eric Gautier. Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 857-871. doi: 10.3934/dcds.2010.26.857

[16]

Shou-Fu Tian. Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval. Communications on Pure & Applied Analysis, 2018, 17 (3) : 923-957. doi: 10.3934/cpaa.2018046

[17]

Roberto A. Capistrano-Filho, Shuming Sun, Bing-Yu Zhang. General boundary value problems of the Korteweg-de Vries equation on a bounded domain. Mathematical Control & Related Fields, 2018, 8 (3&4) : 583-605. doi: 10.3934/mcrf.2018024

[18]

John P. Albert. A uniqueness result for 2-soliton solutions of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3635-3670. doi: 10.3934/dcds.2019149

[19]

Boling Guo, Zhaohui Huo. The global attractor of the damped, forced generalized Korteweg de Vries-Benjamin-Ono equation in $L^2$. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 121-136. doi: 10.3934/dcds.2006.16.121

[20]

Zhaosheng Feng, Yu Huang. Approximate solution of the Burgers-Korteweg-de Vries equation. Communications on Pure & Applied Analysis, 2007, 6 (2) : 429-440. doi: 10.3934/cpaa.2007.6.429

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]