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 $L^2(\T)$ 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.

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 $L^2(\T)$. Dynamics of Partial Differential Equations, 2009, 6 (1), pp.15-34. ⟨hal-00291662v3⟩