A regularity criterion for three-dimensional micropolar fluid equations in Besov spaces of negative regular indices

TL;DR

This paper establishes a new regularity criterion for 3D micropolar fluid equations using Besov space norms of a partial velocity derivative, ensuring smooth solutions under specific integrability conditions.

Contribution

It introduces an improved regularity criterion based on Besov spaces of negative regularity indices, extending previous results for micropolar fluid equations.

Findings

Solutions are smooth if the integral of the Besov norm of a velocity derivative is finite.

The criterion applies to a range of regularity indices, broadening previous conditions.

The results extend and improve existing regularity criteria for micropolar fluids.

Abstract

In this article, we study regularity criteria for the 3D micropolar fluid equations in terms of one partial derivative of the velocity. It is proved that if \begin{equation*} \int^{T}_{0}\|\partial_{3}u\|^{\frac{2}{1-r}}_{\dot{B}^{-r}_{\infty,\infty}} dt<\infty \quad \text{with} \quad 0< r<1, \end{equation*} then, the solutions of the micropolar fluid equations actually are smooth on $(0, T)$. This improves and extends many previous results.