论文标题
在关键$ \ mathbf {l}^Q $基于sobolev和BESOV空间中，BousSinesQ系统的均匀稳定在有限的尺寸内部局部反馈控件中
Uniform stabilization of Boussinesq systems in critical $\mathbf{L}^q$-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls
论文作者
论文摘要
我们考虑在足够光滑的有界域，具有均匀边界条件并受外部来源的情况下定义的D维BousSinesQ系统，该系统被认为会导致不稳定性。流体和热方程式的初始条件都具有低规律性。然后，我们试图通过明确构造的反馈控制，在不稳定的平衡对附近均匀地稳定这种Boussinesq系统，在相应低的规律性空间的临界环境中，这些反馈控制位于任意较小的内部子域中。此外，它们的数量将很小，并且尺寸缩小：更确切地说，对于流体成分的尺寸$（d-1）$，而尺寸为dimension $ 1 $ $ 1 $。所得的稳定度和稳定空间是适合流体速度组件的合适的，紧密的空间（接近$ \ mathbf {l}^3（ω$），对于$ d = 3 $），对于热量组件而言，$ d = 3 $）和热量$ l^q（ω$），$ q> d $ $ q> d $。因此，本文可以看作是\ cite {lpt.1}的扩展，其中相同的内部内部局部均匀稳定结果是通过在同一BESOV设置中使用有限的尺寸反馈控制来实现的。
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 $\mathbf{L}^3(Ω$) for $ d = 3 $) and the space $L^q(Ω$) for the thermal component, $ q > d $. Thus, this paper may be viewed as an extension of \cite{LPT.1}, 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.
