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

Quansen Jiu (School of Mathematical Sciences, Capital Normal University, Beijing, China)

Yanqing Wang (School of Mathematical Sciences, Capital Normal University, Beijing, China)

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.

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