Uniform Stabilization of 3D Navier–Stokes Equations in Low Regularity Besov Spaces with Finite Dimensional, Tangential-Like Boundary, Localized Feedback Controllers
Uniform Stabilization of 3D Navier–Stokes Equations in Low Regularity Besov Spaces with Finite Dimensional, Tangential-Like Boundary, Localized Feedback Controllers
复制标题
DOI：
10.1007/s00205-021-01677-w
复制
发表时间：
2020-11
影响因子：
2.5
通讯作者：
I. Lasiecka;Buddhika Priyasad;R. Triggiani
中科院分区：
文献类型：
--
作者：
I. Lasiecka;Buddhika Priyasad;R. Triggiani
本文通过由控制对组成的“最小”和“最少”侵入反馈策略（Lasiecka和Triggiani in Nonlinear Anal 121：424-446，），对三维Navier-Stokes方程在不稳定平衡解附近的一致稳定化理论中的一个公认的公开问题提供了肯定的解决方案。这里是切向边界反馈控制，作用于边界的任意小部分; u是局部化的内部反馈控制，切向作用于由支持的内部的任意小子集。采取这种理想的策略是不够的。在文献中留下的一个问题是：这样的反馈controlvof pairbe断言是有限维的尺寸也？我们在这里对这个问题给出肯定的回答，从而建立一个最优结果。为了实现所需的有限维反馈切向边界控制，它是必要的，然后在这里放弃希尔伯特-Sobolev功能设置过去的文献，并取代它与一个“正确的”Besov空间设置较低的正则性。这些空间是“接近"的。这种功能设置意义重大。这与最近在非受控N-S方程的全空间中的适定性结果一致（Escauriaza等人，Math Subj Classif 35 K：76 D，1991; Rusin和Sverak，Minimal initial data for potential Navier-Stokes singularities. arXiv：0911.0500; Jia and Šverák in SIAM J Math Anal 45（3）：1448-1459，; Gallagher et al. in Math Ann 355（4）：1527-1559，）.具有紧指标的Besov空间的一个双重关键特征是它们不识别相容性条件，同时具有足够高的拓扑水平来处理适定性和一致稳定性分析中的3d非线性。证明是建设性的，也是“最佳”的切向边界反馈控制器的“最小”数量。新的环境要求解决新的技术和概念问题。这些措施包括建立最大的规律性，直到在所需的适当确定的“正确”的Besov设置的整体闭环线性化问题与切向反馈控制的边界上应用。这一结果也是一个新的贡献，该地区的最大规则的运营商，它适用于纳入了一个边界反馈控制项，而不是齐次边界条件。它避免了直接使用微扰理论。最后，非常稳定的能力，即使是有限维不稳定的投影系统是链接到一个独特的连续属性的一个适当的超定（伴随）Oseen本征问题，这需要存在的内部切向控制作用。
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(Lasiecka and Triggiani in Nonlinear Anal 121:424–446, ). Herevis a tangential boundary feedback control, acting on an arbitrary small partof the boundary;uis a localized, interior feedback control, acting tangentially on an arbitrarily small subsetof the interior supported by. The ideal strategy of takingonis not sufficient. A question left open in the literature is: can such feedback controlvof the pairbe asserted to be finite dimensional also in dimension? 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 controlv, it is here then necessary to abandon the Hilbert-Sobolev functional setting of past literature and replace it with a “right" Besov space setting of lower regularity. These spaces are ‘close’ tofor. This functional setting is significant. It is in line with recent well-posedness results in the full space of the non-controlled N–S equations (Escauriaza et al. in Math Subj Classif 35K:76D, 1991; Rusin and Sverak in Minimal initial data for potential Navier–Stokes singularities. arXiv:0911.0500; Jia and Šverák in SIAM J Math Anal 45(3):1448–1459, ; Gallagher et al. in Math Ann 355(4):1527–1559, ). A double key feature of such Besov spaces with tight indices is that they do not recognize compatibility conditions while having a sufficiently high topological level to handle the 3d-nonlinearity in the analysis of well-posedness and uniform stabilization. The proof is constructive and is “optimal” also regarding the “minimal” number of tangential boundary feedback controllers needed. The new setting requires the solution of novel technical and conceptual issues. These include establishing maximal regularity up toin the required suitably identified “right" Besov setting for the overall closed-loop linearized problem with tangential feedback control applied on the boundary. This result is also a new contribution to the area of maximal regularity as the operator to which it applies incorporates a boundary feedback control term rather than homogeneous boundary conditions. It escapes direct use of perturbation theory. Finally, the very ability to stabilize even the finite dimensional unstable projected system is linked to a Unique Continuation Property of a suitably over-determined (adjoint) Oseen eigenproblem, which requires the presence of the interior tangential-like controluacting on.