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