, Volume 12, Issue 2, pp 355-379

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

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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.

Submitted: April 2001, Revised: August 2001.