# 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
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