APPROXIMATION OF EXPONENTIAL ORDER OF THE ATTRACTOR OF A TURBULENT-FLOW

In this work, we consider the two-dimensional Navier-Stokes equations with periodic boundary conditions, describing the evolution of homogeneous turbulent flows. We use the method of Debussche and Temam [A. Debussche and R. Temam, convergent families of approximate inertial manifolds, J. Math. Pures Appl., to appear] to construct approximate inertial manifolds (AIMs) whose order decreases exponentially fast with respect to the dimension of the manifold. We recall that an AIM is a smooth manifold of solutions M such that the attractor A is contained in a neighborhood of M of thinness eta, eta is called the order of the manifold. The dependence of all the constants with respect to physical parameters, especially the Grashof number, is made explicit