Abstract In this paper, we consider the long time behavior of solutions for 3D incompressible MHD equations with fractional Laplacian. Firstly, in a periodic bounded domain, we prove the existence of a global attractor. The analysis reveals a relation between the Laplacian exponent and the regularity of the spaces of velocity and magnetic fields. Finally, in the whole space ℝ 3 {\mathbb{R}^{3}} , we establish the sharp algebraic decay rate of solutions to the generalized MHD system provided that the parameters satisfy α , β ∈ ( 0 , 2 ] {\alpha,\beta\in(0,2]} .

中文翻译：

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3D 广义 MHD 方程解的长时间行为

摘要 在本文中，我们考虑了具有分数拉普拉斯算子的 3D 不可压缩 MHD 方程解的长时间行为。首先，在周期性有界域中，我们证明了全局吸引子的存在。分析揭示了拉普拉斯指数与速度和磁场空间的规律性之间的关系。最后，在整个空间 ℝ 3 {\mathbb{R}^{3}} 中，如果参数满足 α , β ∈ ( 0 , 2 ] {\ alpha,\beta\in(0,2]} .