向列型液晶流体动力学中的胶体均匀化
Colloidal Homogenisation for the Hydrodynamics of Nematic Liquid Crystals
AI总结：
本文分析研究了向列型液晶胶体流体动力学模型，通过均匀化方法揭示速度场满足Darcy方程、指向矢遵循梯度流的解耦现象，并引入新的Aubin-Lions引理变体解决扩展算子缺失问题。
摘要：
本文从分析角度探索了向列型液晶胶体流体动力学的简化模型。我们将速度场的Stokes方程与指向矢场的Ginzburg-Landau输运热流相耦合。研究聚焦于一个有界空间域，其中包含周期性分布的胶体颗粒，这些颗粒对向列型液晶施加无锚定条件。通过逐步将颗粒尺寸减小至零并同时增加颗粒数量，我们深入研究了相关的均匀化问题。我们的分析揭示了一种解耦形式：速度场渐近地满足与指向矢无关的Darcy方程，而指向矢则遵循不受速度场影响的梯度流。均匀化过程中最复杂的方面之一是指向矢场缺乏扩展算子来保持与系统能量相关的均匀估计。我们通过一种新颖的Aubin-Lions引理变体解决了这一挑战，该变体专门针对均匀化问题进行了调整。
英文摘要：
This paper analytically explores a simplified model for the hydrodynamics of nematic liquid crystal colloids. We integrate a Stokes equation for the velocity field with a Ginzburg-Landau transported heat flow for the director field. The study focuses on a bounded spatial domain containing periodically distributed colloidal particles, which impose no-anchoring conditions on the nematic liquid crystal. By progressively reducing the particle size to zero and simultaneously increasing the number of particles, we delve into the associated homogenisation problem. Our analysis uncovers a form of decoupling where the velocity field asymptotically satisfies a Darcy equation, independent of the director, while the director follows a gradient flow, unaffected by the velocity field. One of the most intricate aspects of the homogenisation process is the absence of an extension operator for the director field that preserves the uniform estimates related to the system's energy. We address this challenge with a novel variation of the Aubin-Lions lemma, specifically adapted for homogenisation problems.
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