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 \(\alpha \ge \frac{5}{4}\), \(\beta \ge 0\) and \(\alpha +\beta \ge \frac{7}{4}\), 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.

中文翻译：

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三维分数阶微极性方程的整体正则性

三维（3D）不可压缩微极性方程的光滑解的整体适定性是一个难题。本文重点研究分数耗散\（（-\ Delta）^ {\ alpha} u \）和\（（-\ Delta）^ {\ beta} w \）的3D不可压缩微极性方程。我们的目标是建立具有最小耗散量的分数微极性方程的全局正则性。我们证明，如果\（\ alpha \ ge \ frac {5} {4} \），\（\ beta \ ge 0 \）和\（\ alpha + \ beta \ ge \ frac {7} {4} \ ），分数阶3D微极性方程对于任何足够平滑的数据始终拥有唯一的全局经典解。另外，我们还获得了3D微极性方程的整体正则性，其傅立叶乘法器给出的耗散对数弱于分数拉普拉斯算子。