论文标题
3D Navier-Stokes方程的均匀稳定在关键的BESOV空间中，具有有限的尺寸，切向边界，局部反馈控制器
Uniform stabilization of 3D Navier-Stokes equations in critical Besov spaces with finite dimensional, tangential-like boundary, localized feedback controllers
论文作者
论文摘要
本文通过通过“最小值”和“最小”的“入侵反馈策略，由控制对$ \ \ \ \ \ \ cyite contecte cans over of稳定的均衡解决方案，在不稳定平衡解决方案的附近统一稳定理论中为公认的公开问题提供了解决方案。这里$ v $是一个切向边界反馈控制，作用于边界$γ$的任意小部分$ \widetildeγ$； $ u $是一种本地化的内部反馈控制，在$ \widetildeγ$支持的内部任意小子集$ω$上切线。在$ω$上服用$ u = 0 $的理想策略是不够的。文献中留下的一个问题是：这对$ \ {v，u \} $的这种反馈控制$ v $是否可以在尺寸$ d = 3 $中被认为是有限维度？我们在这里对这个问题给出了肯定的答案，从而确立了最佳结果。为了实现反馈切向边界控制$ v $的所需有限维度，在这里有必要放弃希尔伯特过去文学的设置，并用besov设置替换为$ l^3（ω）$ for $ d = 3 $。它与非控制N-S方程的整个空间中最近的关键适应性相符。此类BESOV空间的一个关键特征是它们不识别兼容性条件。证明是建设性的，并且对于所需的切向边界控制器的“最小”数量也是“最佳”。新设置需要在所需的关键BESOV设置中确定最大规律性，以确保整体闭环线性化问题，并在边界上应用切向反馈控制。最后，切向边界动作的最小量与过度确定的Oseen Eigenproblems的唯一延续问题有关。
The present paper provides a solution in the affirmative to a recognized open problem in the theory of uniform stabilization of 3-dimensional Navier-Stokes equations in the vicinity of an unstable equilibrium solution, by means of a `minimal' and `least' invasive feedback strategy which consists of a control pair $\{ v,u \}$ \cite{LT2:2015}. Here $v$ is a tangential boundary feedback control, acting on an arbitrary small part $\widetildeΓ$ of the boundary $Γ$; while $u$ is a localized, interior feedback control, acting tangentially on an arbitrarily small subset $ω$ of the interior supported by $\widetildeΓ$. The ideal strategy of taking $u = 0$ on $ω$ is not sufficient. A question left open in the literature was: Can such feedback control $v$ of the pair $\{ v, u \}$ be asserted to be finite dimensional also in the dimension $d = 3$? We here give an affirmative answer to this question, thus establishing an optimal result. To achieve the desired finite dimensionality of the feedback tangential boundary control $v$, it is here then necessary to abandon the Hilbert setting of past literature and replace it with a Besov setting which are `close' to $L^3(Ω)$ for $d=3$. It is in line with recent critical well-posedness in the full space of the non-controlled N-S equations. A key feature of such Besov spaces with tight indices is that they do not recognize compatibility conditions. The proof is constructive and is "optimal" also regarding the "minimal" number of tangential boundary controllers needed. The new setting requires establishing maximal regularity in the required critical Besov setting for the overall closed-loop linearized problem with tangential feedback control applied on the boundary. Finally, the minimal amount of tangential boundary action is linked to the issue of unique continuation of over-determined Oseen eigenproblems.
