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A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation. (English) Zbl 0999.35069
Summary: We prove that for the Navier-Stokes equation $\frac{\partial u}{\partial t}+ u\cdot\nabla u+\nabla p= -(-\Delta)^\alpha u,$ with dissipation $$(-\Delta)^\alpha$$ where $$1 < \alpha < 5 /4$$, and smooth initial data, the Hausdorff dimension of the singular set at time of first blow-up is at most $$5 - 4 \alpha$$. This unifies two directions from which one might approach the problem of global solvability, though it provides no direct progress on either.

##### MSC:
 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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