This paper focuses on the 3D incompressible magnetohydrodynamic (MHD) equations with mixed partial dissipation and magnetic diffusion. Our main result assesses the global stability of perturbations near the steady solution given by a background magnetic field. The stability problem on the MHD equations with partial or no dissipation has attracted considerable interests recently and there are substantial developments. The new stability result presented here is among the very few stability conclusions currently available for ideal or partially dissipated MHD equations. As a special consequence of the techniques introduced in this paper, we obtain the small data global well-posedness for the 3D incompressible Navier-Stokes equations without vertical dissipation.

中文翻译：

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具有混合局部耗散和接近平衡的磁扩散的 3D 不可压缩 MHD 系统的全局解

本文重点研究具有混合局部耗散和磁扩散的 3D 不可压缩磁流体动力学 (MHD) 方程。我们的主要结果评估了背景磁场给出的稳定解附近扰动的全局稳定性。具有部分耗散或无耗散的 MHD 方程的稳定性问题最近引起了相当大的兴趣，并且取得了实质性的进展。此处提出的新稳定性结果是目前可用于理想或部分耗散 MHD 方程的极少数稳定性结论之一。作为本文介绍的技术的特殊结果，我们获得了无垂直耗散的 3D 不可压缩 Navier-Stokes 方程的小数据全局适定性。