ON THE EXISTENCE OF SLOW MANIFOLDS FOR PROBLEMS WITH DIFFERENT TIMESCALES

We consider time dependent systems of partial differential equations ( PDE) whose solutions can vary on two different timescales. An example is given by the Navier-Stokes equations for slightly compressible flow s. By proper initialization, the fast timescale can be suppressed to a ny given order; however, this does generally not imply the existence o f a slow manifold. Since the PDE solutions are uniformly smooth in spa ce, one can approximate the PDE system by a finite dimensional Galerki n system. Under suitable assumptions, this finite dimensional dynamica l system will have a slow manifold.