Buoyancy-driven fluids such as many atmospheric and oceanic flows and the Rayleigh–Bénard convection are modeled by the Boussinesq systems. By rigorously estimating the large-time behavior of solutions to a special Boussinesq system, this paper reveals a fascinating phenomenon on buoyancy-driven fluids that the temperature can actually stabilize the fluids. The Boussinesq system concerned here governs the motion of perturbations near the hydrostatic equilibrium. When the buoyancy forcing is not present, the velocity of the fluid obeys the 2D Navier–Stokes equation with only vertical dissipation and its Sobolev norm could potentially grow even though its precise large-time behavior remains open. This paper shows that the temperature through the coupling and interaction tames and regularizes the fluids, and causes the velocity (measured in Sobolev norms) to decay in time. Optimal decay rates are obtained.

Boussinesq系统模拟了浮力驱动的流体，例如许多大气和海洋流动以及Rayleigh-Bénard对流。通过严格估计特殊Boussinesq系统的解的长时间行为，本文揭示了浮力驱动流体的一种引人入胜的现象，即温度实际上可以稳定流体。这里涉及的Boussinesq系统控制流体静力学平衡附近的摄动运动。当不存在浮力时，流体的速度服从仅具有垂直耗散的二维Navier-Stokes方程，并且即使其精确的长时间行为仍然开放，其Sobolev范数也可能会增长。本文表明，温度通过耦合和相互作用而驯服并使流体均匀化，并导致速度（以Sobolev规范衡量）随时间衰减。获得最佳衰减率。