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

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

2008-06-27

arxiv.org/abs/0806.4578v3

Dynamics of Partial Differential Equations 6, 1 (2009) 15-34

Mathematics

Analysis of PDEs

Corrected version. To appear in Dynamics of Partial Differential Equations

Scientific paper

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.

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