We consider the 3D incompressible hypodissipative Navier-Stokes equations, when the dissipation is given as a fractional Laplacian (-Delta)(s) for s is an element of(3/4, 1), and we provide a new bootstrapping scheme that makes it possible to analyse weak solutions locally in space-time. This includes several homogeneous Kato-Ponce type commutator estimates which we localize in space, and which seems applicable to other parabolic systems with fractional dissipation. We also provide a new estimate on the pressure, parallel to(-Delta)(s)p parallel to(H1)less than or similar to parallel to(-Delta)(s/2)u parallel to(2)(L2). We apply our main result to prove that any suitable weak solution u satisfies del(n)u is an element of L-loc(p,infinity)(R(3)x(0, infinity)) for p = 2(3s-1)/n+2s-1, n = 1, 2. As a corollary of our local regularity theorem, we improve the partial regularity result of Tang-Yu (2015) 26, and obtain an estimate on the box-counting dimension of the singular set S, d(B)(S boolean AND{t >= t(0)}) 0. (C) 2022 Elsevier Inc. All rights reserved.
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