Nematic–Isotropic Phase Transition in Liquid Crystals: A Variational Derivation of Effective Geometric Motions
Nematic–Isotropic Phase Transition in Liquid Crystals: A Variational Derivation of Effective Geometric Motions
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DOI：
10.1007/s00205-021-01681-0
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发表时间：
2020-10
影响因子：
2.5
通讯作者：
Tim Laux;Yuning Liu
中科院分区：
文献类型：
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作者：
Tim Laux;Yuning Liu
本文利用由控制对组成的“最小”和“最小”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镇定理论中一个公认的开放问题给出了肯定的解。这是一个切向边界反馈控制，作用于边界的任意小部分；是一种局部的内部反馈控制，切向地作用于由支持的任意小的内部子集。理想的接管策略是不够的。文献中留下的一个问题是：是否可以断言这种对的反馈控制在维度上也是有限维的？我们在这里对这个问题给出一个肯定的答案，从而建立一个最优结果。为了获得期望的反馈切向边界控制的有限维数，这里有必要放弃过去文献中的Hilbert-Sobolev函数设置，代之以正则性较低的“右”Besov空间设置。这些空间是“接近”的。这个功能设置很重要。这与最近在非受控N-S方程全空间的适定性结果一致（Escauriaza et al. in Math Subj Classif 35K:76D, 1991; Rusin and Sverak in Minimal initial data for potential Navier-Stokes singularity）。arXiv: 0911.0500;数学学报，45(3):1448-1459；Gallagher等人在数学Ann 355(4): 1527-1559,)。这类紧指标Besov空间的一个双重关键特征是，它们不识别相容条件，同时在适定性和均匀镇定分析中具有足够高的拓扑水平来处理三维非线性。这个证明是建设性的，也是“最优”的，关于切向边界反馈控制器所需的“最小”数量。新的环境要求解决新的技术和概念问题。这些包括在边界上应用切向反馈控制的整体闭环线性化问题中建立所需的适当识别的“正确”Besov设置的最大正则性。这个结果也是对最大正则性领域的一个新贡献，因为它所应用的算子包含了一个边界反馈控制项，而不是齐次边界条件。它逃避了微扰理论的直接使用。最后，稳定有限维不稳定投影系统的能力与一个适当的超定（伴随）Oseen特征问题的唯一延拓性质有关，该问题要求存在内部类切控制。
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.