In this paper, we investigate the n-dimensional incompressible magnetohydrodynamic (MHD) equations with fractional dissipation and magnetic diffusion. Firstly, employing energy methods, we demonstrate that if the initial data is sufficiently small in \(H^s(\mathbb {R}^n)\) with \(s=1+\frac{n}{2}-2\alpha ~(0<\alpha <1)\), then the system possesses a global solution. In order to establish the uniqueness, we enhance the regularity of the initial data and prove that if \((u_0,b_0)\) is small in \(H^s(\mathbb {R}^n)\) with \(s=1+\frac{n}{2}-\alpha ~(0<\alpha <1)\), then the system admits a unique global solution. Secondly, by applying frequency decomposition, we obtain \(\Vert u,b\Vert _{L^2}\rightarrow 0,~t\rightarrow \infty \). Assuming in addition that the initial data \(u_0,b_0\in L^p(1\le p<2)\), we establish optimal decay estimates for the solutions and their higher order derivatives by employing a more refined frequency decomposition approach. In the case \(\alpha = 0\), the system corresponds to a damped MHD equations, which have been previously investigated in [34]. Our results improve ones in [34] by extending the solution space from \(H^s(s>\frac{n}{2}+1)\) to \(B^s_{2,1}(s\ge \frac{n}{2}+1)\).

在本文中，我们研究了具有分数耗散和磁扩散的 n 维不可压缩磁流体动力学 (MHD) 方程。首先，采用能量方法，我们证明如果\(H^s(\mathbb {R}^n)\)中的初始数据足够小，且\(s=1+\frac{n}{2}-2 \alpha ~(0<\alpha <1)\)，则系统拥有全局解。为了建立唯一性，我们增强了初始数据的规律性，并证明如果\((u_0,b_0)\)在\(H^s(\mathbb {R}^n)\)中较小，则用\( s=1+\frac{n}{2}-\alpha ~(0<\alpha <1)\)，则系统承认唯一的全局解。其次，通过应用频率分解，我们得到\(\Vert u,b\Vert _{L^2}\rightarrow 0,~t\rightarrow \infty \)。另外假设初始数据\(u_0,b_0\in L^p(1\le p<2)\)，我们通过采用更精细的频率分解方法为解及其高阶导数建立最佳衰减估计。在\(\alpha = 0\)的情况下，系统对应于阻尼 MHD 方程，该方程先前已在 [34] 中进行过研究。我们的结果通过将解空间从\(H^s(s>\frac{n}{2}+1)\)扩展到\(B^s_{2,1}(s\ge ) 来改进[34]中的结果\frac{n}{2}+1)\)。