Well-posedness and dynamics for the fractional Ginzburg-Landau equation

Abstract

This article studies the global well-posedness and long-time dynamics for the nonlinear complex Ginzburg–Landau equation involving fractional Laplacian. The global existence and some uniqueness criterion of weak solutions are given with compactness method. To study the strong solutions with the semigroup method, we generalize some pointwise estimates for the fractional Laplacian to the complex background and study carefully the linear evolution of the equation. Finally, the existence of global attractors is studied.

Keywords:

• Ginzburg–Landau equation
• fractional Laplacian
• weak solutions
• smooth solutions
• global attractors

AMS Subject Classification:

• 35Qxx

Acknowledgements

Supported by the NNSF of China under Grant No. 11001285.