This paper studies the global regularity problem for the 2D micropolar Rayleigh-B e ̵́ nard convection system with velocity zero dissipation, micro-rotation velocity Laplace dissipation and temperature critical dissipation. By introducing a combined quantity and using the technique of Littlewood-Paley decomposition, we establish the global regularity result of solutions to this system.

In this paper, we derive two regularity criteria of solutions to the nematic liquid crystal flows. More precisely, we prove the local smooth solution $(u, d)$ is regular if and only if one of the following two conditions is satisfied: (i) $\nabla_{h} u_{h}\in L^{\frac{2p}{2p-3}}(0,T; L^{p}(\mathbb{R}^{3})),\ \partial_{3} d\in L^{\frac{2q}{q-3}}(0,T; L^{q}(\mathbb{R}^{3})),\ \frac{3}{2}< p\leq\infty,\ 3< q\leq\infty$; and (ii) $\nabla_{h} u_{h}\in L^{q}(0,T; L^{p}(\mathbb{R}^{3})),\ \frac{3}{p}+\frac{2}{q}\leq 1, \ 3