We consider the d-dimensional Boussinesq system defined on a sufficiently smooth bounded domain, with homogeneous boundary conditions, and subject to external sources, assumed to cause instability. The initial conditions for both fluid and heat equations are taken of low regularity. We then seek to uniformly stabilize such Boussinesq system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of explicitly constructed, feedback controls, which are localized on an arbitrarily small interior subdomain. In addition, they will be minimal in number, and of reduced dimension: more precisely, they will be of dimension $ (d-1) $ for the fluid component and of dimension $ 1 $ for the heat component. The resulting space of well-posedness and stabilization is a suitable, tight Besov space for the fluid velocity component (close to $ \mathbf{L}^3(\Omega $) for $ d = 3 $) and the space $ L^q(\Omega $) for the thermal component, $ q > d $. Thus, this paper may be viewed as an extension of [63], where the same interior localized uniform stabilization outcome was achieved by use of finite dimensional feedback controls for the Navier-Stokes equations, in the same Besov setting.

中文翻译：

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Boussinesq系统在关键阶段的均匀稳定 \ begin {document} $ \ mathbf {L} ^ q $ \ end {document}有限维内部局部反馈控制的基于Sobolev和Besov的空间

我们考虑在足够光滑的有界域上定义的d维Boussinesq系统，具有均匀的边界条件，并受外部因素的影响，并假定会引起不稳定性。流体方程和热方程的初始条件均具有较低的规律性。然后，我们寻求通过明确构造的反馈控制（位于局部较小的内部子域上），在相应的低规则空间的临界环境中，在不稳定的平衡对附近均匀稳定此类Boussinesq系统。另外，它们的数量将最小并且尺寸将减小：更精确地，对于流体成分，它们将具有尺寸（d-1）$，对于热成分，它们将具有尺寸$ 1 $。由此产生的适当摆放和稳定的空间是合适的，流体速度分量的紧Besov空间（对于$ d = 3 $，接近$ \ mathbf {L} ^ 3（\ Omega $））和热分量的空间$ L ^ q（\ Omega $），q > d $。因此，本文可被视为[63]，其中在相同的Besov设置中，通过对Navier-Stokes方程使用有限维反馈控制来实现相同的内部局部均匀稳定结果。