Global Regularity of the Three-Dimensional Fractional Micropolar Equations

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.