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

Abstract

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 $\alpha \ge 5/6$.

24 pages. We improve the results of the first version. We obtain the optimal dimensional Hausdorff estimate of possible time singular points of weak solutions to the fractional Navier-Stokes equations

References in corpus (1)

• On the regularity of a class of generalized quasi-geostrophic equations