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On regularity criteria for weak solutions to the micropolar fluid equations in Lorentz space. (English) Zbl 1191.35217
Summary: The regularity of weak solutions and blow-up criteria for smooth solutions to the micropolar fluid equations in three dimensional space are studied in the Lorentz space ${L}^{p,\infty }\left({ℝ}^{3}\right)$. We obtain that if $u\in {L}^{q}\left(0,T;{L}^{p,\infty }\left({ℝ}^{3}\right)\right)$ for $\frac{2}{q}+\frac{3}{p}\le 1$ with $3, or if $\nabla u\in {L}^{q}\left(0,T;{L}^{p,\infty }\left({ℝ}^{3}\right)\right)$ for $\frac{2}{q}+\frac{3}{p}\le 2$ with $\frac{3}{2}, or if the pressure $P\in {L}^{q}\left(0,T;{L}^{p,\infty }\left({ℝ}^{3}\right)\right)$ for $\frac{2}{q}+\frac{3}{p}\le 2$ with $\frac{3}{2}, or if $\nabla P\in {L}^{q}\left(0,T;{L}^{p,\infty }\left({ℝ}^{3}\right)\right)$ for $\frac{2}{q}+\frac{3}{p}\le 3$ with $1, then the weak solution $\left(u,\omega \right)$ satisfying the energy inequality is a smooth solution on $\left[0,T\right)$.
##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76D03 Existence, uniqueness, and regularity theory 35B44 Blow-up (PDE) 76W05 Magnetohydrodynamics and electrohydrodynamics