Local regularity of weak solutions of the hypodissipative Navier-Stokes equations

We consider the 3D incompressible hypodissipative Navier-Stokes equations, when the dissipation is given as a fractional Laplacian (−Δ)s for s∈(34,1), 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, ‖(−Δ)sp‖H1≲‖(−Δ)s2u‖L22. We apply our main result to prove that any suitable weak solution u satisfies ∇nu∈Llocp,∞(R3×(0,∞)) 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, dB(S∩{t≥t0})≤13(15−2s−8s2) for every t0>0.