A new sharp estimate on the dimension of the attractor for the Dirichlet problem of the complex Ginzburg-Landau equation

TL;DR

This paper provides a sharper estimate for the dimension of the global attractor in the complex Ginzburg-Landau equation with Dirichlet boundary conditions, leveraging improved eigenvalue bounds.

Contribution

It introduces a new, sharper estimate for the attractor's dimension by applying an enhanced eigenvalue lower bound to the Dirichlet Laplacian.

Findings

Sharper upper bound on attractor dimension

Improved understanding of complex Ginzburg-Landau dynamics

Enhanced eigenvalue estimate application

Abstract

Using the improved lower bound on the sum of the eigenvalues of the Dirichlet Laplacian proved by A. D. Melas (Proc. Amer. Math. Soc. \textbf{131} (2003) 631-636), we report a new and sharp estimate for the dimension of the global attractor associated to the complex Ginzburg-Landau equation supplemented with Dirichlet boundary conditions.