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

Stabilizing effect of large average initial velocity in forced dissipative PDEs invariant with respect to Galilean transformations