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

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 the 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 viscous 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 the existence of a locally attracting solution.