This paper focuses on the stability problem for perturbations near a hydrostatic equilibrium associated with the three-dimensional Boussinesq equations with partial dissipation. We mainly study the global stability and large-time behavior of solutions to this system with only horizontal dissipation and obtain three results. The first establishes the global \(H^2\)-stability, the second obtains the large-time asymptotic of the solutions, and the last assesses the asymptotic linear stability with explicit decay rates. Finally, we remark that these results can also be extended to the system with mixed partial dissipation.

中文翻译：

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具有部分耗散的三维Boussinesq方程在静液压平衡附近的稳定性

本文重点研究与带有部分耗散的三维Boussinesq方程相关的静水平衡附近的摄动稳定性问题。我们主要研究仅具有水平耗散的该系统解的整体稳定性和长时间行为，并获得三个结果。第一个建立全局\（H ^ 2 \） -稳定性，第二个建立解的长时间渐近性，最后一个建立具有明确衰减率的渐近线性稳定性。最后，我们指出，这些结果还可以扩展到具有混合部分耗散的系统。