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

Abstract:

In this paper 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 $L^{p,\infty }(\mathbb {R}^3)$. We obtain that if $u\in L^q(0,T;L^{p,\infty }(\mathbb {R}^3))$ for $\frac 2q+\frac 3p\le 1$ with $3<p\le \infty$ or if $\nabla u\in L^q(0,T;L^{p,\infty }(\mathbb {R}^3))$ for $\frac 2q+\frac 3p\le 2$ with $\frac 32<p\le \infty$ or if the pressure $P\in L^q(0,T;L^{p,\infty }(\mathbb {R}^3))$ for $\frac 2q+\frac 3p\le 2$ with $\frac 32<p\le \infty$ or if $\nabla P\in L^q(0,T;L^{p,\infty }(\mathbb {R}^3))$ for $\frac 2q+\frac 3p\le 3$ with $1<p\le \infty$, then the weak solution $(u,\omega )$ satisfying the energy inequality is a smooth solution on $[0,T)$.