A cheap Caffarelli—Kohn—Nirenberg inequality for the Navier—Stokes equation with hyper-dissipation

Abstract.

We prove that for the Navier—Stokes equation with dissipation $(-\Delta)^\alpha$ where 1 < α < 5 /4, and smooth initial data, the Hausdorff dimension of the singular set at time of first blow up is at most 5 — 4α. This unifies two directions from which one might approach the problem of global solvability, though it provides no direct progress on either.