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Zbl 1191.35217
Yuan, Baoquan
On regularity criteria for weak solutions to the micropolar fluid equations in Lorentz space. (English)
[J] Proc. Am. Math. Soc. 138, No. 6, 2025-2036 (2010). ISSN 0002-9939; ISSN 1088-6826/e

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 $L^{p,\infty}(\Bbb R^3)$. We obtain that if $u\in L^q(0,T;L^{p,\infty}(\Bbb R^3))$ for $\frac 2q+\frac 3p\le 1$ with $3<p\le\infty$, or if $\nabla u\in L^q(0,T;L^{p,\infty}(\Bbb R^3))$ for $\frac 2q+ \frac 3p\le 2$ with $\frac 32<p\le\infty$, or if the pressure $P\in L^q(0,T;L^{p,\infty}(\Bbb R^3))$ for $\frac 2q+ \frac 3p\le 2$ with $\frac 32<p\le\infty$, or if $\nabla P\in L^q(0,T;L^{p,\infty}(\Bbb R^3))$ 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 $[0,T)$.

MSC 2000:: *35Q35 Other equations arising in fluid mechanics
76D03 Existence, uniqueness, and regularity theory
35B44
76W05 Flows in presence of electromagnetic forces

Keywords: micropolar fluid equations; regularity of weak solutions; Lorentz spaces

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