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

The forced and weakly damped Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. Starting from L2 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 Hs, s ∈ (0; 1), whose radius depends only on s, the L2 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 Hs. In addition we prove that higher order Sobolev norms are bounded for all positive times.