Partial regularity and $L^3$-norm concentration effects around possible blow-up points for the micropolar fluid equations

Abstract

The micropolar fluid system is a model based on the Navier-Stokes equations which considers two coupled variables: the velocity field $\vec{u}$ and the microrotation field $\vec{\ddot{I}}‰$. Assuming an additional condition over the variable $\vec{u}$ we will first prove that weak solutions $(\vec{u},\vec{\ddot{I}}‰)$ of this system are smooth. Then, we will present a concentration effect of the $L_x^3$ norm of the velocity field $\vec{u}$ near a possible singular time.