This paper is concerned with the global regularity problem on the micropolar Rayleigh-Bénard problem with only velocity dissipation in $\mathbb {R}^{d}$ with $d=2\ or\ 3$. By fully exploiting the special structure of the system, introducing two combined quantities and using the technique of Littlewood-Paley decomposition, we establish the global regularity of solutions to this system in $\mathbb {R}^{2}$. Moreover, we obtain the global regularity for fractional hyperviscosity case in $\mathbb {R}^{3}$ by employing various techniques including energy methods, the regularization of generalized heat operators on the Fourier frequency localized functions and logarithmic Sobolev interpolation inequalities.

本文关注在$\mathbb {R}^{d}$中$d=2\ 或 \3$中仅有速度耗散的微极 Rayleigh-Bénard 问题的全局规律性问题。通过充分利用系统的特殊结构，引入两个组合量，利用Littlewood-Paley分解技术，我们建立了该系统在$\mathbb {R}^{2}$中解的全局正则性。此外，我们通过采用各种技术，包括能量方法、傅里叶频率局部函数上广义热算子的正则化和对数 Sobolev 插值不等式，获得了$\mathbb {R}^{3}$中分数高粘度情况的全局正则性。