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 $L^{p,\infty}(\mathbb{R}^3)$. We obtain that if $u\in L^q(0,T;L^{p,\infty}(\mathbb{R}^3))$ for $\frac2q+\frac3p\le 1$ with $3<p\le \infty$; or $\nabla u\in L^q(0,T;L^{p,\infty}(\mathbb{R}^3))$ for $\frac2q+\frac3p\le 2$ with $\frac32<p\le \infty$; 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 $1<p\le \infty$, then the weak solution $(u,\omega)$ satisfying the energy inequality is a smooth solution on $[0,T)$.