Global Attractor for Weakly Damped Forced KdV Equation in Low Regularity on T

TL;DR

This paper investigates the long-term dynamics of the weakly damped, forced KdV equation on a periodic domain, establishing boundedness and the existence of a global attractor in low regularity Sobolev spaces.

Contribution

It proves uniform boundedness of solutions and the existence of a global attractor in negative Sobolev spaces for the first time.

Findings

Solutions are uniformly bounded in $ ext{H}^s$ for $s > -rac{1}{2}$.

Existence of a global attractor in $ ext{H}^s$ for $s > -rac{1}{2}$.

The attractor is compact in $ ext{H}^{s+3}$.

Abstract

In this paper we consider the long time behavior of the weakly damped, forced Korteweg-de Vries equation in the Sololev spaces of the negative indices in the periodic case. We prove that the solutions are uniformly bounded in $\dot{H}^s(\T)$ for $s>-\dfrac{1}{2}$. Moreover, we show that the solution-map possesses a global attractor in $\dot{H}^s(\T)$ for $s>-\dfrac{1}{2}$, which is a compact set in $H^{s+3}(\T)$.