Finite Reductions for Dissipative Systems and Viscous Fluid-Dynamic Models on ${\mathbb T}^2$

We reconsider the reduction method introduced for Hamiltonian systems by Amann, Conley and Zehnder. We propose an extension of these techniques to evolutive PDE systems of dissipative type and prove that, under suitable regularity conditions, a finite number of spectral modes controls exactly the time evolution of the complete problem. The problem of finite reduction for a two-dimensional modified Navier–Stokes equations is considered and an estimate of the dimension of the reduced space is given, valid for any time t > 0. Comparison is made with the asymptotic finite dimension that has been obtained for the true Navier–Stokes equations.