Abstract This paper focuses on a special two-dimensional (2D) Boussinesq system modeling buoyancy driven fluids. It governs the motion of the velocity and temperature perturbations near the hydrostatic balance. This is a partially dissipated system with the velocity involving only the vertical dissipation. We are able to establish the global stability and the large-time behavior of the solutions. In particular, our results reveal that the buoyancy force actually stabilizes the fluids through the coupling and interaction. Without the coupling, the 2D Navier-Stokes equation with only vertical dissipation is not known to be stable. Mathematically the coupling allows us to deduce that both the velocity and the temperature obey degenerate damped wave equations, which generates the stabilization effect.

中文翻译：

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具有部分耗散的二维 Boussinesq 方程的稳定性和大时间行为

摘要 本文着重于一种特殊的二维 (2D) Boussinesq 系统建模浮力驱动流体。它控制流体静力平衡附近的速度和温度扰动的运动。这是一个部分耗散系统，其速度仅涉及垂直耗散。我们能够建立解决方案的全局稳定性和长时间行为。特别是，我们的结果表明，浮力实际上通过耦合和相互作用稳定了流体。如果没有耦合，只有垂直耗散的 2D Navier-Stokes 方程是不稳定的。从数学上讲，耦合使我们能够推断出速度和温度都服从退化阻尼波动方程，从而产生稳定效应。