Global attractor and asymptotic smoothing effects for the weakly damped cubic Schr\"odinger equation in $L^2(\T)$

TL;DR

This paper demonstrates that the weakly damped cubic Schrödinger equation in $L^2(\T)$ has a global attractor and exhibits asymptotic smoothing, with the attractor being compact in $H^2(\T)$, revealing optimal regularity properties.

Contribution

The paper establishes the existence of a global attractor for the weakly damped cubic Schrödinger flow in $L^2(\T)$ and proves its asymptotic smoothing into $H^2(\T)$, extending previous results on flow-map behavior.

Findings

Existence of a global attractor in $L^2(\T)$.

Attractor is compact in $H^2(\T)$.

Asymptotic smoothing effect is optimal.

Abstract

We prove that the weakly damped cubic Schr\"odinger flow in $L^2(\T)$ provides a dynamical system that possesses a global attractor. The proof relies on a sharp study of the behavior of the associated flow-map with respect to the weak $ L^2(\T) $-convergence inspired by a previous work of the author. Combining the compactness in $ L^2(\T) $ of the attractor with the approach developed by Goubet, we show that the attractor is actually a compact set of $ H^2(\T) $. This asymptotic smoothing effect is optimal in view of the regularity of the steady states.