Dynamics of Partial Differential Equations
Volume 6 (2009)
Global attractor and asymptotic smoothing effects for the weakly damped cubic Schrödinger equation in L²(T)
Pages: 15 – 34
We prove that the weakly damped cubic Schrödinger flow in L² (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 L² (T)-convergence inspired by . Combining the compactness in L² (T) of the attractor with the approach developed in , we show that the attractor is actually a compact set of H² (T). This asymptotic smoothing effect is optimal in view of the regularity of the steady states.
global attractor, asymptotic smoothing, cubic Schrödinger flow
2010 Mathematics Subject Classification