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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: Baoquan Yuan
Journal: Proc. Amer. Math. Soc. 138 (2010), 2025-2036
MSC (2010): Primary 35Q35, 76D03
Published electronically: January 27, 2010
MathSciNet review: 2596038
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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 $ \frac2q+\frac3p\le 1$ with $ 3<p\le \infty$ or if $ \nabla u\in L^q(0,T;L^{p,\infty}(\mathbb{R}^3))$ for $ \frac2q+\frac3p\le 2$ with $ \frac32<p\le \infty$ or if 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 if $ \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)$.


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Additional Information

Baoquan Yuan
Affiliation: School of Mathematics and Information Science, Henan Polytechnic University, Henan 454000, People’s Republic of China
Email: bqyuan@hpu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10232-9
PII: S 0002-9939(10)10232-9
Keywords: Micropolar fluid equations, regularity of weak solutions, Lorentz spaces.
Received by editor(s): August 20, 2009
Published electronically: January 27, 2010
Additional Notes: The author was partially supported by the National Natural Science Foundation of China (grant No. 10771052), the Program for Science & Technology Innovation Talents in Universities of Henan Province (grant No. 2009HASTIT007), the Doctor Fund of Henan Polytechnic University (grant No. B2008-62), and the Innovation Scientists and Technicians Troop Construction Projects of Henan Province.
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.