# Global regularity of the three-dimensional fractional micropolar equations

## TL;DR

This paper proves the global regularity of 3D fractional micropolar equations under minimal dissipation conditions, advancing understanding of their well-posedness for smooth initial data.

## Contribution

It establishes the global existence and uniqueness of solutions for fractional micropolar equations with minimal dissipation assumptions, including logarithmically weaker Fourier multiplier dissipations.

## Key findings

- Global regularity for $oldsymbol{eta ext{ and } eta}$ under specified conditions.
- Unique classical solutions exist for smooth initial data.
- Extension to Fourier multipliers weaker than fractional Laplacian.

## Abstract

The global well-posedness of the smooth solution to the three-dimensional (3D) incompressible micropolar equations is a difficult open problem. This paper focuses on the 3D incompressible micropolar equations with fractional dissipations $( \Delta)^{\alpha}u$ and $(-\Delta)^{\beta}w$.Our objective is to establish the global regularity of the fractional micropolar equations with the minimal amount of dissipations. We prove that, if $\alpha\geq \frac{5}{4}$, $\beta\geq 0$ and $\alpha+\beta\geq\frac{7}{4}$, the fractional 3D micropolar equations always possess a unique global classical solution for any sufficiently smooth data. In addition, we also obtain the global regularity of the 3D micropolar equations with the dissipations given by Fourier multipliers that are logarithmically weaker than the fractional Laplacian.