Exponential Attractor for Complex Ginzburg-Landau Equation in Three-dimensions

Abstract

" In this paper, we consider the complex Ginzburg-Landau equation (CGL) in three spatial dimensions u_t = ρu + (1 + iϒ\r\n)Δu - (1 + iμ) |u|^{2σ} u, \\qquad(1) u(0, x) = u_0(x), \\qquad(2) where u is an unknown complex-value function defined in 3+1 dimensional space-time R^{3+1}, Δ is a Laplacian in R\u00b3, ρ > 0, \r\nϒ, μ are real parameters. Ω ∈ R\u00b3 is a bounded domain. We show that the semigroup S(t) associated with the problem (1), (2) satisfies Lipschitz continuity and the squeezing property for the initial-value problem (1), (2) with the initial-value condition belonging to H²(Ω ), therefore we obtain the existence of exponential attractor."