Degenerate Pullback Attractors for the 3D Navier-Stokes Equations

As was found in Cheskidov and Kavlie (Pullback attractors for generalized evolutionary systems. DCDS-B 20(3), 749-779, 2015), the 3D Navier-Stokes equations with a translationally bounded force possesses pullback attractors A(w)(t) in a weak sense. Moreover, those attractors consist of complete bounded trajectories. In this paper, we present a sufficient condition under which the pullback attractors are degenerate. That is, if the Grashof number is small enough, each section of the pullback attractor is a single point on a unique, complete, bounded, strong solution. We then apply our results to provide a new proof of the existence of a unique, strong, periodic solution to the 3D Navier-Stokes with a small, periodic forcing term.