The micropolar equations model a class of fluids with nonsymmetric stress tensor such as fluids consisting of particles suspended in a viscous medium. While the viscous micropolar equations are globally well-posed, their inviscid counterparts remain an open problem. This makes it both natural and compelling to investigate the two-dimensional micropolar equations with general fractional dissipation \((-\Delta )^{\alpha }u\) and \((-\Delta )^{\beta }w\) and to explore how much dissipation is required to ensure global well-posedness. The fractional dissipation allows us to simultaneously treat a family of equations including those with the standard Laplacian dissipation and is relevant in some physical circumstances. The global regularity results of the two borderline cases \(\alpha =1, \beta =0\) and \(\alpha =0, \beta =1\) have been proved. However, the global regularity of the situation for the general critical case \(\alpha +\beta =1\) with \(0<\alpha <1\) appears to be out of reach. Actually, when the dissipation is split among the equations, the dissipation is no longer as efficient as in the borderline cases. Even for the subcritical case \(\alpha +\beta >1\) with \(\alpha ,\,\beta \in (0,\,1)\) , the corresponding global regularity problem is not trivial and has not been completely resolved yet. One of the main ideas is to fully exploit the structure of the system and control the vorticity via the evolution equation of a combined quantity of the vorticity and the micro-rotation angular velocity. Then, by maximally exploiting the nonlinear lower bounds for the fractional Laplacian operator and the DeGiorgi-Nash estimate, we obtain the sharpest global regularity result in the subcritical case with the minimal amount of dissipation. Therefore, compared with the previous works, our result is more closer to the general critical case.
