Dynamics of Partial Differential Equations

Volume 11 (2014)

Number 1

Regularity of attractor for 3D derivative Ginzburg-Landau equation

Pages: 89 – 108

DOI: https://dx.doi.org/10.4310/DPDE.2014.v11.n1.a5

Authors

Shujuan Lü (School of Mathematics and Systems Science, Beihang University, Beijing, China)

Zhaosheng Feng (Department of Mathematics, University of Texas-Pan American, Edinburg, Texas, U.S.A.)

Abstract

In this paper, we are concerned with a three-dimensional derivative Ginzburg-Landau equation with a periodic initial value condition. The smoothing property of the solution is established by a uniform priori estimates. The existence of the global attractors, $\mathcal{A}_i \subset H^i_p (\Omega) (i=2,3,\dots)$ for the semi-group ${ \{ S^{(i)} (t) \} }_{t \geq 0}$ of the operators generated by the equation is proved by using the compactness principle. Finally, the regularity of the global attractors, namely, $\mathcal{A}_2 = \mathcal{A}_3 = \dots = \mathcal{A}_m$, is proved by using the method of semi-group decomposition.

Keywords

Ginzburg-Landau equation, global attractor, regularity, Hölder inequality, priori estimates, semi-group decomposition

2010 Mathematics Subject Classification

35B41, 35B45, 35Q56

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Published 8 April 2014