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

TL;DR

This paper investigates the regularity of solutions to the micropolar fluid equations and demonstrates an $L^3$-norm concentration effect near potential blow-up points, extending understanding of singularity formation.

Contribution

It proves smoothness of weak solutions under certain conditions and reveals $L^3$-norm concentration effects around possible singularities in micropolar fluids.

Findings

Weak solutions become smooth under additional conditions.

Identifies $L^3$-norm concentration near potential blow-up points.

Provides insights into singularity formation in micropolar fluids.

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 $\vec\omega$. Assuming an additional condition over the variable $\vec u$ we will first prove that weak solutions $(\vec u, \vec\omega)$ of this system are smooth. Then, we will present a concentration effect of the $L^3_x$ norm of the velocity field $\vec u$ near a possible singular time.