Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2D Navier-Stokes equations

The fractal and Hausdorff dimensions of the universal attractor for the Navier-Stokes equations are studied in two space dimensions. The functional setting is recalled and the precise definition of the universal attractor is given. The equations governing the transport of finite-dimensional volume elements under the action of the Navier-Stokes system are established, and global Liapunov exponents are defined and studied. The existence of a critical dimension which enjoys the property that every N-dimensional volume element in the phase space decays exponentially over time is proved. Estimates for this critical dimension are given in terms of the Grashof number, separately for the periodic and aperiodic cases. It is proved that the expressions given by Kaplan and Yorke (1979), when combined with the introduced global Liapunov exponents, yield upper bounds for the fractal and Hausdorff dimensions of the universal attractor.