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
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show all references

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Comm. Pure Appl. Math., 64 (2011), 591-648. doi: 10.1002/cpa.20356.  Google Scholar

[2]

Discrete Contin. Dyn. Syst., 10 (2003), 31-52. doi: 10.3934/dcds.2004.10.31.  Google Scholar

[3]

GAFA, 3 (1993), 209-262. doi: 10.1007/BF01895688.  Google Scholar

[4]

Physica D, 192 (2004), 265-278 doi: 10.1016/j.physd.2004.01.023.  Google Scholar

[5]

J. Amer. Math. Soc., 16 (2003), 705-749. 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. Functional Analysis, 151 (1997), 384-436. doi: 10.1006/jfan.1997.3148.  Google Scholar

[8]

J. Diff. Eqs., 74 (1988), 369-390. doi: 10.1016/0022-0396(88)90010-1.  Google Scholar

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J. Diff. Eqs., 110 (1994), 356-359. doi: 10.1006/jdeq.1994.1071.  Google Scholar

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[11]

Duke Math. J., 135 (2006), 327-360. doi: 10.1215/S0012-7094-06-13524-X.  Google Scholar

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[17]

Nonlinear Differ. Equ. Appl., 18 (2011), 273-285. doi: 10.1007/s00030-010-0095-9.  Google Scholar

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