Global attractor and asymptotic smoothing effects for the weakly damped cubic Schr\"odinger equation in $L^2(\text{T})$

We prove that the weakly damped cubic Schr"odinger flow in $L^2(\text{T})$ 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 $L^2(\text{T})$-convergence inspired by a previous work of the author. Combining the compactness in $L^2(\text{T})$ of the attractor with the approach developed by Goubet, we show that the attractor is actually a compact set of $H^2(\text{T})$. This asymptotic smoothing effect is optimal in view of the regularity of the steady states.