# American Institute of Mathematical Sciences

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:
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##### 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
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