Attractors of derivative complex Ginzburg-Landau equation in unbounded domains

Abstract

The Ginzburg-Landau-type complex equations are simpli?ed mathematical models for various pattern formation systems in mechanics, physics, and chemistry. In this paper, the derivative complex Ginzburg-Landau (DCGL) equation in an unbounded domain Ω? R² is studied. We extend the Gagliardo-Nirenberg inequality to the weighted Sobolev spaces introduced by S. V. Zelik. Applied this Gagliardo-Nirenberg inequality of the weighted Sobolev spaces and based on the technique for the semi-linear system of parabolic equations which has been developed by M. A. Efendiev and S. V. Zelik, the global attractor A in the corresponding phase space is constructed, the upper bound of its Kolmogorov s ε-entropy is obtained, and the spatial chaos of the attractor A for DCGL equation in R² is detailed studied.