This paper concerns the Cauchy problem of the three-dimensional nonhomogeneous incompressible magnetohydrodynamic (MHD) equations with density-dependent viscosity and vacuum. We first establish some key a priori algebraic decay-in-time rates of the strong solutions. Then after using these estimates, we also obtain the global existence and large time asymptotic behavior of strong solutions in the whole three-dimensional space, provided that the initial velocity and magnetic field are suitable small in the \(\dot{H}^{\beta }\)-norm for some \(\beta \in (1/2, 1]\). Note that any smallness and compatibility conditions assumed on the initial data are not used in this result. Moreover, the density can contain vacuum states and even have compact support initially.

本文涉及具有密度依赖的粘度和真空的三维非均匀不可压缩磁流体动力学（MHD）方程的柯西问题。我们首先建立强解的关键先验代数随时间的衰减率。然后，使用这些估计值后，只要初始速度和磁场在\（\ dot {H} ^ {较小的情况下适合）较小，我们还可以获得整个三维空间中强解的整体存在性和大时间渐近行为。\ beta} \） -某些\（\ beta \ in（1/2，1] \）的范数。请注意，此结果中未使用任何假设于初始数据上的小和相容性条件，而且密度可以包含处于真空状态，甚至最初具有紧凑的支撑。