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

Luc Molinet (1)

1 LAGA - Laboratoire Analyse, Géométrie et Applications

We prove that the weakly damped cubic Schrödinger flow in $L^2(\mathbb{T})$ 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 \cite{L}. Combining the compactness in $ L^2(\T) $ of the attractor with the approach developed in \cite{G}, 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.

Mots clés: global attractor; asymptotic smoothing effect; Schrödinger equation

Domaines: Equations aux dérivées partielles [math.AP]

Luc Molinet. Global attractor and asymptotic smoothing effects for the weakly damped cubic Schrödinger equation in $L^2(\T)$. 2008. ⟨hal-00291662v1⟩