This paper is devoted to the stability and decay estimates of solutions to the two-dimensional magneto-micropolar fluid equations with partial dissipation. Firstly, focus on the 2D magneto-micropolar equation with only velocity dissipation and partial magnetic diffusion, we obtain the global existence of solutions with small initial in \(H^s({\mathbb {R}}^2)\) \((s>1)\), and by fully exploiting the special structure of the system and using the Fourier splitting methods, we establish the large time decay rates of solutions. Secondly, when the magnetic field has partial dissipation, we show the global existence of solutions with small initial data in \(\dot{B}^0_{2,1}({\mathbb {R}}^2)\). In addition, we explore the decay rates of these global solutions are correspondingly established in \(\dot{B}^m_{2,1}({\mathbb {R}}^2)\) with \(0 \le m \le s\), when the initial data belongs to the negative Sobolev space \(\dot{H}^{-l}({\mathbb {R}}^2)\) (for each \(0 \le l <1\)).

本文致力于具有部分耗散的二维磁微极流体方程解的稳定性和衰减估计。首先，关注仅具有速度耗散和部分磁扩散的二维磁微极方程，我们得到了\(H^s({\mathbb {R}}^2)\) \) 中小初始解的全局存在性(s>1)\)，并充分利用系统的特殊结构，利用傅立叶分裂方法，建立了解的大时间衰减率。其次，当磁场发生部分耗散时，我们在\(\dot{B}^0_{2,1}({\mathbb {R}}^2)\)中证明了具有小初始数据的解的全局存在性。此外，我们探索了这些全局解的衰减率相应地在\(\dot{B}^m_{2,1}({\mathbb {R}}^2)\)中建立，其中\(0 \le m \le s\)，当初始数据属于负 Sobolev 空间\(\dot{H}^{-l}({\mathbb {R}}^2)\)（对于每个\(0 \le l <1\) )。