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Global regularity for the micropolar Rayleigh-Bénard problem with only velocity dissipation

Part of: Incompressible inviscid fluids Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  26 August 2021

Lihua Deng  and
Haifeng Shang
Lihua Deng
Affiliation:
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454000, People's Republic of China (denglihua@hpu.edu.cn, hfshang@163.com)
Haifeng Shang
Affiliation:
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454000, People's Republic of China (denglihua@hpu.edu.cn, hfshang@163.com)
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Abstract

This paper is concerned with the global regularity problem on the micropolar Rayleigh-Bénard problem with only velocity dissipation in $\mathbb {R}^{d}$ with $d=2\ or\ 3$. By fully exploiting the special structure of the system, introducing two combined quantities and using the technique of Littlewood-Paley decomposition, we establish the global regularity of solutions to this system in $\mathbb {R}^{2}$. Moreover, we obtain the global regularity for fractional hyperviscosity case in $\mathbb {R}^{3}$ by employing various techniques including energy methods, the regularization of generalized heat operators on the Fourier frequency localized functions and logarithmic Sobolev interpolation inequalities.

Keywords

Micropolar Rayleigh-Bénard problempartial dissipationglobal regularity

MSC classification

Primary: 35Q35: PDEs in connection with fluid mechanics
Secondary: 35B65: Smoothness and regularity of solutions 76B03: Existence, uniqueness, and regularity theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 5 , October 2022 , pp. 1109 - 1138
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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