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 = \u03c1u + (1 + i\u03d2\r\n)\u0001\u0394u - (1 + i\u03bc) |u|^{2\u03c3} 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}, \u0394 is a Laplacian in R\u00b3, \u03c1 > 0, \r\n\u03d2, \u03bc are real parameters. \u03a9 \u2208 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 H\u00b2(\u03a9 ), therefore we obtain the existence of exponential attractor."