We consider several direct and adjoint Boussinesq static problems under different types of over-determined conditions. We then conclude, in each case, that the solution pair corresponding to {fluid velocity, scalar temperature} must vanish identically on the whole domain, so that the pressure is then constant (Unique Continuation Property). In going from the direct to the adjoint problem, the coupling operators between the fluid and the thermal equations switch places. As a result, the adjoint Boussinesq system has a more favorable structure than the direct Boussinesq system and hence yields UCP results under weaker requirements; typically, a reduction by one or even two units on the number of components of the fluid vector being involved in the assumptions. To illustrate: in the key direct Boussinesq problem, over-determination consists of the additional vanishing of the solution pair in a common arbitrarily small subset of the interior. In contrast, in the corresponding adjoint Boussinesq problem, only the first \((d-1)\) components of the d-dimensional fluid velocity vector need to be assumed as vanishing on the interior subset. These UCPs for the adjoint problem are critical ingredients in the solution of corresponding uniform stabilization problems of (direct) dynamic Boussinesq systems by suitable finite dimensional feedback controls. They allow one to verify a corresponding Kalman algebraic condition for controllability.

我们考虑了在不同类型的超定条件下的一些直接的和伴随的Boussinesq静态问题。然后，我们得出结论：在每种情况下，与{流体速度，标量温度}对应的溶液对必须在整个域上完全消失，因此压力是恒定的（唯一连续性）。在从直接问题转向伴随问题时，流体方程和热方程之间的耦合算子会切换位置。结果，伴随的Boussinesq系统比直接的Boussinesq系统具有更好的结构，因此在较弱的要求下会产生UCP结果。通常，假设中涉及的流体矢量的分量数量减少一个或什至两个单位。举例说明：在直接的Boussinesq问题中，过度确定包括在内部任意小的公共子集中对溶液对的额外消失。相反，在相应的伴随Boussinesq问题中，只有第一个d维流体速度矢量的\（（d-1）\）分量需要假定为在内部子集中消失。通过适当的有限维反馈控制，这些伴随问题的UCP是解决（直接）动态Boussinesq系统相应均匀稳定问题的关键要素。他们允许人们验证相应的卡尔曼代数条件的可控制性。