Stability and exponential decay for the 2D magneto-micropolar equations with partial dissipation

Abstract

The stability and large-time behavior problem on the magneto-micropolar equations has evoked a considerable interest in recent years. In this paper, we study the stability and exponential decay near magnetic hydrostatic equilibrium to the two-dimensional magneto-micropolar equations with partial dissipation in the domain $\mathrm{\Omega }=\mathbb{T}\times \mathbb{R}$. In particular, we takes advantage of the geometry of the domain $\mathbb{T}\times \mathbb{R}$ to divide u into zeroth mode and the nonzero modes, and obey a strong version of the Poincaré's inequality, which plays a crucial role in controlling the nonlinearity. Moreover, we find that the oscillation part of the solution decays exponentially to zero. Finally, our result mathematically verifies that the stabilization effect of a background magnetic field on magneto-micropolar fluids.

Keywords:

• Magneto-micropolar fluids
• partial dissipation
• stability
• large-time behavior

2020 MSC:

• 35A01
• 35B35
• 76E25

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research of Weiwei Wang was supported by the Natural Science Foundation of Fujian Province of China (Grant No. 2022J01105).