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

TL;DR

This paper analyzes the long-term behavior of solutions to the forced and weakly damped KdV equation on the torus, showing decomposition into decaying linear and smoother nonlinear parts, and establishing the existence and size of a global attractor.

Contribution

It provides a new proof for the existence of a smooth global attractor and quantifies its size in Sobolev spaces for the forced, damped KdV equation.

Findings

Solutions decompose into linear decay and smoother nonlinear parts

All solutions are attracted to a bounded set in H^s

Quantitative bounds on the attractor size depending on parameters

Abstract

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 $H^s$.