  We describe a topological method to study the dynamics of dissipative PDEs on
a torus with rapidly oscillating forcing terms. We show that a dissipative PDE,
which is invariant with respect to Galilean transformations, with a large
average initial velocity can be reduced to a problem with rapidly oscillating
forcing terms. We apply the technique to the Burgers equation, and the
incompressible 2D Navier-Stokes equations with a time-dependent forcing. We
prove that for a large initial average speed the equation admits a bounded
eternal solution, which attracts all other solutions forward in time. For the
incompressible 3D Navier-Stokes equations we establish existence of a locally
attracting solution.
