Finite fractal dimension of the global attractor for a class of non-Newtonian fluids

We present a new criterion of finiteness of the fractal dimension of the attractor via the method of short trajectories developed in [1]. As an application, we deal with the so-called generalized Navier-Stokes equations characterized by nonlinear polynomial dependence of (p - 1) order between the stress tensor and the symmetric velocity gradient. We study the case p greater than or equal to 2 subject to space-periodic boundary conditions. The existence of the global attractor with finite fractal dimension is then obtained in the following cases: (i) in two dimensions if p greater than or equal to 2, and (ii) in three dimensions if p greater than or equal to 11/5. (C) 1999 Elsevier Science Ltd. All rights reserved.