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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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