Dynamics of Partial Differential Equations

Volume 6 (2009)

Number 1

Global attractor and asymptotic smoothing effects for the weakly damped cubic Schrödinger equation in L²(T)

Pages: 15 – 34

DOI: https://dx.doi.org/10.4310/DPDE.2009.v6.n1.a2

Author

Luc Molinet (L.A.G.A., Institut Galilée, Université Paris-Nord, Villetaneuse, France)

Abstract

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 [18]. Combining the compactness in L² (T) of the attractor with the approach developed in [10], 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.

Keywords

global attractor, asymptotic smoothing, cubic Schrödinger flow

2010 Mathematics Subject Classification

35-xx

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Published 1 January 2009