Higher derivatives of suitable 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 $(-\Delta )^s$ for $s\in (\frac34,1)$, and prove that any suitable weak solutions $u$ satisfies $\nabla^n u \in L^{p,\infty }_{\text{loc}}(\mathbb{R}^3\times(0,\infty))$ for $p=\frac{2(3s-1)}{n+2s-1}$, $n=1,2$. Our main approach is inspired by the work of Vasseur (2010). The central point of our analysis is a new bootstrapping argument that enables us to efficiently localize the equation and study higher derivatives. This includes several homogeneous Kato-Ponce type commutator estimates, and seems applicable to other parabolic systems with fractional dissipation. We also provide a new estimate on the pressure, $\|(-\Delta)^s p \|_{\mathcal{H}^1}\lesssim \| (-\Delta )^{\frac s2} u \|^2_{L^2}$. As a corollary of our local regularity theorem, we improve the partial regularity result of Tang and Yu (2015), and obtain an estimate on the box-counting dimension of the singular set $S$, $d_B(S\cap \{t\geq t_0 \} )\leq \frac13 (15-2s-8s^2) $ for every $t_0>0$.