Global regularity and time decay for the 2D magneto-micropolar system with fractional dissipation and partial magnetic diffusion

This paper focuses on the 2D incompressible magneto-micropolar system with the kinematic dissipation given by the fractional operator $(-\Delta)^\alpha$, the magnetic diffusion by partial Laplacian and the spin dissipation by the fractional operator $(-\Delta)^\gamma$. We prove that this system, with any $0 \lt \alpha \lt \gamma \lt 1$ and $\alpha + \gamma \gt 1$, always possesses a unique global smooth solution $(\mathbf{u}, \mathbf{b}, \mathrm{w}) \in H^s (\mathbb{R}^2) (s \geq 3)$ if the initial data is sufficiently smooth. In addition, we study the large-time behavior of these smooth solutions and obtain optimal large-time decay rates.

Keywords

magneto-micropolar system, fractional operator, partial dissipation, large-time decay

2010 Mathematics Subject Classification

35B40, 35B65, 76D03

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The author is supported by NSFC Grant No. 11701049, Sichuan Youth Science Technology Foundation (2014JQ0003) and China Scholarship Council Fund (201808510059) and Panzhihua University Foundation (035200075).

Received 10 November 2019

Received revised 12 August 2021

Accepted 13 November 2021

Published 26 May 2022