This paper develops an effective approach to establishing the optimal decay estimates on solutions of the 3D anisotropic magnetohydrodynamic (MHD) equations with only horizontal dissipation. As our first step, we prove the global existence and stability of solutions to the aforementioned MHD system emanating from any initial data with small \(H^1\)-norm. Due to the lack of dissipation in the vertical direction, the large-time behavior does not follow from the classical approaches. The analysis of the nonlinear terms are much more difficult than in the case of full dissipation. In particular, we need to represent the MHD equations in an integral form, exploit cancellations and other properties such as the incompressibility in order to control terms involving vertical derivatives.

本文开发了一种有效方法，可以对仅具有水平耗散的 3D 各向异性磁流体动力学 (MHD) 方程的解建立最佳衰减估计。作为我们的第一步，我们证明了上述 MHD 系统解的全局存在性和稳定性，这些解源自任何具有小\(H^1\)范数的初始数据。由于缺乏垂直方向的耗散，大时间行为并不遵循经典方法。非线性项的分析比全耗散情况要困难得多。特别是，我们需要以积分形式表示 MHD 方程，利用抵消和其他性质（例如不可压缩性）来控制涉及垂直导数的项。