A note on the regularity criterion for the micropolar fluid equations in homogeneous Besov spaces

Abstract

This paper gives a further investigation on the regularity criteria for three-dimensional micropolar equations in Besov spaces. More precisely, it is proved that the weak solution $(u, \omega)$ is regular if the velocity $u$ satisfies

$$\int_{0}^{T}\| \nabla_{h}u_{h}\|_{\dot{B}_{p,\frac{2p}{3}}^{0}}^{q} d t<\infty,\ with\ \ \frac{3}{p}+\frac{2}{q}=2,\ \frac{3}{2}<p\leq\infty,$$
or $$\int_{0}^{T}\| \nabla_{h}u\|_{\dot{B}_{\infty ,\infty}^{-1}}^{\frac{8}{3}} d t<\infty,$$
or $$\int_{0}^{T}\|\nabla_{h} u_{h}\|_{\dot{B}_{\infty,\infty}^{-\alpha}}^{\frac{2}{2-\alpha}} d t<\infty,\ with\ 0< \alpha< 1. $$