Global regularity for d-dimensional micropolar equations with fractional dissipation

ABSTRACT

This paper is devoted to the global in time existence of classical solutions to the d-Dimensional (dD) micropolar equations with fractional dissipation. Micropolar equations model a class of fluids with nonsymmetric stress tensor such as fluids consisting of particles suspended in a viscous medium. It remains unknown whether or not smooth solutions of the classical 3D micropolar equations can develop finite-time singularities. The purpose here is to explore the global regularity of solutions for dD micropolar equations under the smallest amount of dissipation. We establish the global regularity for two important fractional dissipation cases. Direct energy estimates are not sufficient to obtain the desired global a priori bounds in each case. To overcome the difficulties, we employ the Besov space techniques.

KEYWORDS:

• Micropolar equations
• fractional dissipation
• global regularity

AMS SUBJECT CLASSIFICATIONS:

• 35Q35
• 35B65
• 76B03

Acknowledgements

The authors would like to thank the Editor and the anonymous referees for handling this manuscript and their useful suggestions, which improve the presentation of the manuscript.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by NSFC [grant number 11201124], [grant number 11471103]; Foundation for University Key Teacher by the Henan Province [grant number 2015GGJS-070]; Outstanding Youth Foundation of Henan Polytechnic University [grant number J2014-03].