A note on the regularity criterion of weak solutions for the micropolar fluid equations

Abstract

The aim of this paper is to investigate the regularity criterion of
Leray-Hopf weak solutions to the 3D incompressible micropolar fluid
equations. It is shown that if
\begin{equation*}
\int_{0}^{T}\frac{\left\Vert \nabla \pi (t)\right\Vert _{L^{r}}^{\frac{2r}{3(r-1)}}}{\left\Vert u(\cdot ,t)\right\Vert _{L^{3}}^{\alpha }+\left\Vert
\omega (\cdot ,t)\right\Vert _{L^{3}}^{\alpha }}dt<\infty \text{ \ \ with \ }
\alpha =\left\{
\begin{array}{c}
3,\text{ \ \ }1<r\leq \frac{9}{7}, \\
\frac{2r}{3(r-1)},\text{ \ \ }\frac{9}{7}<r<3,
\end{array}
\right.
\end{equation*}
then the corresponding weak solution $(u,\omega )$ is regular on $[0,T]$,
which is an obvious extension of the previous results.