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# Dynamics of Partial Differential Equations

## Volume 11 (2014)

### Number 4

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

Pages: 321 – 343

DOI: http://dx.doi.org/10.4310/DPDE.2014.v11.n4.a2

#### Authors

#### 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 $5/6 \leq \alpha \lt 5/4$. 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 $\alpha \geq 5/6$.

#### Keywords

Navier-Stokes equations, fractional dissipation, weak solutions, Hausdorff dimension, eventual regularity

#### 2010 Mathematics Subject Classification

Primary 35-xx. Secondary 76-xx.