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

### MSC:

 35Q35 PDEs in connection with fluid mechanics 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 35B44 Blow-up in context of PDEs 76W05 Magnetohydrodynamics and electrohydrodynamics
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