Global attractor and asymptotic smoothing effects for the weakly damped cubic Schrödinger equation in $L^2(\T)$
Abstract
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2008
- DOI:
- 10.48550/arXiv.0806.4578
- arXiv:
- arXiv:0806.4578
- Bibcode:
- 2008arXiv0806.4578M
- Keywords:
-
- Mathematics - Analysis of PDEs;
- 35Q55;
- 35B41
- E-Print:
- Corrected version. To appear in Dynamics of Partial Differential Equations