Asymptotic numerical analysis for the Navier-Stokes equations, 1

Our aim in this work is to show that, in a 'permanent regime', the behavior of a viscous incompressible fluid can be, in principle, determined by the study of a finite number of modes. It is proved that the behavior for t yields infinity of the solution to the Navier-Stokes equations is completely determined by its projection on appropriate finite dimensional subspaces, corresponding to eigenspaces of the linear operator, or more general subspaces, including finite element subspaces. Some indications on the dimension of such subspaces are given.