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Regularity of weak solutions to magneto-micropolar fluid equations. (English) Zbl 1240.35421
Summary: In this article, we study the regularity of weak solutions and the blow-up criteria for smooth solutions to the magneto-micropolar fluid equations in ${ℝ}^{3}$. We obtain the classical blow-up criteria for smooth solutions $\left(u,\omega ,b\right)$, i.e., $u\in {L}^{q}\left(0,T;{L}^{p}\left({ℝ}^{3}\right)\right)$ for $\frac{2}{q}+\frac{3}{p}\le 1$ with $3 or $\nabla u\in {L}^{q}\left(0,T;{L}^{p}\right)$ for $\frac{3}{2} satisfying $\frac{2}{q}+\frac{3}{p}\le 2$. Moreover, our results indicate that the regularity of weak solutions is dominated by the velocity $u$ of the fluid. In the end-point case $p=\infty$, the blow-up criteria can be extended to more general spaces $\nabla u\in {L}^{1}\left(0,T;{\stackrel{˙}{B}}_{\infty ,\infty }^{0}\left({ℝ}^{3}\right)\right)$.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35B65 Smoothness and regularity of solutions of PDE 35D30 Weak solutions of PDE 35B44 Blow-up (PDE) 76W05 Magnetohydrodynamics and electrohydrodynamics
##### Keywords:
magneto-micropolar fluid equations; regularity; blow-up