On the regularity criteria of weak solutions to the micropolar fluid equations in Lorentz space

In this paper the regularity of weak solutions and the blow-up criteria of smooth solutions to the micropolar fluid equations on three dimension space are studied in the Lorentz space $L^{p,\mathrm{\infty }}({\mathbb{R}}^3)$. We obtain that if $u\in L^q(0,T;L^{p,\mathrm{\infty }}({\mathbb{R}}^3))$ for $\frac{2}{q}+\frac{3}{p}\le 1$ with $3<p\le \mathrm{\infty }$; or $\mathrm{\nabla }u\in L^q(0,T;L^{p,\mathrm{\infty }}({\mathbb{R}}^3))$ for $\frac{2}{q}+\frac{3}{p}\le 2$ with $\frac{3}{2}<p\le \mathrm{\infty }$; or the pressure $P\in L^q(0,T;L^{p,\infty}($\mathbb{R}$^3))$for$\frac2q+\frac3p\le 2$with$\frac32<p\le \infty$;or$\nabla P\in L^q(0,T;L^{p,\infty}($\mathbb{R}$^3))$for$\frac2q+\frac3p\le 3$with$1<p\le \infty$,thentheweaksolution$(u,\omega)$satisfyingtheenergyinequalityisasmoothsolutionon$[0,T)$.