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

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

2011-08-16

arxiv.org/abs/1108.3358v1

Mathematics

Analysis of PDEs

18 pages

Scientific paper

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 $H^s$.

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