# American Institute of Mathematical Sciences

August  2018, 38(8): 4133-4162. doi: 10.3934/dcds.2018180

## Global regularity for the 2D micropolar equations with fractional dissipation

 1 College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China 2 School of Mathematical Science, Anhui University, Hefei 230601, China 3 Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078, USA 4 School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China 5 Department of Mathematics and Statistics, Jiangsu Normal University, 101 Shanghai Road, Xuzhou 221116, Jiangsu, China

* Corresponding author: Zhuan Ye

Received  December 2017 Revised  March 2018 Published  May 2018

Fund Project: Dong was partially supported by the National Natural Science Foundation of China (NO. 11571240; No.11271019) and Research Fund of Shenzhen University (No. 2017056). Wu was partially supported by NSF grants DMS 1209153 and DMS 1624146, by the AT & T Foundation at Oklahoma State University (OSU) and by NSFC (No.11471103, a grant awarded to Professor Baoquan Yuan). Xu was partially supported by the National Natural Science Foundation of China (No. 11371059; No. 11471220; No. 11771045). Ye was supported by the Foundation of Jiangsu Normal University (No. 16XLR029), the Natural Science Foundation of Jiangsu Province (No. BK20170224) and the National Natural Science Foundation of China (No. 11701232).

Micropolar equations, modeling micropolar fluid flows, consist of coupled equations obeyed by the evolution of the velocity $u$ and that of the microrotation $w$. This paper focuses on the two-dimensional micropolar equations with the fractional dissipation $(-Δ)^{α} u$ and $(-Δ)^{β}w$, where $0<α, β<1$. The goal here is the global regularity of the fractional micropolar equations with minimal fractional dissipation. Recent efforts have resolved the two borderline cases $α = 1$, $β = 0$ and $α = 0$, $β = 1$. However, the situation for the general critical case $α+β = 1$ with $0<α<1$ is far more complex and the global regularity appears to be out of reach. When the dissipation is split among the equations, the dissipation is no longer as efficient as in the borderline cases and different ranges of $α$ and $β$ require different estimates and tools. We aim at the subcritical case $\alpha+\beta>1$ and divide $\alpha\in (0,1)$ into five sub-intervals to seek the best estimates so that we can impose the minimal requirements on $\alpha$ and $\beta$. The proof of the global regularity relies on the introduction of combined quantities, sharp lower bounds for the fractional dissipation and delicate upper bounds for the nonlinearity and associated commutators.

Citation: Bo-Qing Dong, Jiahong Wu, Xiaojing Xu, Zhuan Ye. Global regularity for the 2D micropolar equations with fractional dissipation. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4133-4162. doi: 10.3934/dcds.2018180
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Regularity region
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