On possible time singular points and eventual regularity of weak solutions to the fractional Navier-Stokes equations

In this paper, we intend to reveal how the fractional dissipation $(-\Delta)^{\alpha}$ affects the regularity of weak solutions to the 3d generalized Navier-Stokes equations. Precisely, it will be shown that the $(5-4\alpha)/2\alpha$ dimensional Hausdorff measure of possible time singular points of weak solutions on the interval $(0,\infty)$ is zero when $5/6\le\alpha< 5/4$. To this end, the eventual regularity for the weak solutions is firstly established in the same range of $\alpha$. It is worth noting that when the dissipation index $\alpha$ varies from $5/6$ to $5/4$, the corresponding Hausdorff dimension is from $1$ to $0$. Hence, it seems that the Hausdorff dimension obtained is optimal. Our results rely on the fact that the space $H^{\alpha}$ is the critical space or subcritical space to this system when $\alpha\geq5/6$.