Regularity Criterion for Weak Solution to the 3D Micropolar Fluid Equations

Abstract

Regularity criterion for the 3D micropolar fluid equations is investigated. We prove that, for some $T>0$, if ${{\displaystyle \int }}_0^T\parallel v_{x_3}{\parallel }_{L^{\varrho }}^{\rho }dt<\infty$, where $3/\varrho +2/\rho \le 1$ and $\varrho \ge 3$, then the solution $(v,w)$ can be extended smoothly beyond $t=T$. The derivative $v_{x_3}$ can be substituted with any directional derivative of v.