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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 $(-Δ)^α$ affects the regularity of weak solutions to the 3d generalized Navier-Stokes equations. Precisely, it will be shown that the $(5-4α)/2α$ dimensional Hausdorff measure of possible time singular points of weak solutions on the interval $(0,\infty)$ is zero when $5/6\leα< 5/4$. To this end, the eventual regularity for the weak solutions is firstly established in the same range of $α$. It is worth noting that when the dissipation index $α$ 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 $H^α$ is the critical space or subcritical space to this system when $α\geq5/6$.