Strong Well-Posedness of the Q-Tensor Model for Liquid Crystals: The Case of Arbitrary Ratio of Tumbling and Aligning Effects \(\xi \)

IF 2.6 1区 数学 Q1 MATHEMATICS, APPLIED
Matthias Hieber, Amru Hussein, Marc Wrona
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引用次数: 0

Abstract

The Beris–Edwards model of nematic liquid crystals couples an equation for the molecular orientation described by the Q-tensor with a Navier–Stokes type equation with an additional non-Newtonian stress caused by the molecular orientation. Both equations contain a parameter \(\xi \in \mathbb {R}\) measuring the ratio of tumbling and alignment effects. Previous well-posedness results largely vary on the space dimension n and the constraints of the parameter \(\xi \in \mathbb {R}\). This work addresses strong well-posedness of this model, first locally and then globally for small initial data, both in the \(L^p\)-\(L^2\)-setting for \(p > \frac{4}{4-n}\), in the general cases, i.e., for \(n = 2, 3\) and without any restriction on \(\xi \). The approach is based on methods from quasilinear equations and the fact that the associated linearized operator admits maximal \(L^p\)-\(L^2\)-regularity. The proof of the latter property relies on techniques from sectorial operators, Schur complements and \(\mathcal {J}\)-symmetry.

液晶 Q 张量模型的强好拟性:任意比例的翻滚效应和对齐效应的情况 $$\xi $$
向列液晶的 Beris-Edwards 模型将 Q 张量描述的分子取向方程与纳维-斯托克斯方程耦合在一起,后者带有由分子取向引起的附加非牛顿应力。这两个方程都包含一个参数 \(\xi \in \mathbb {R}\),用于测量翻滚效应和排列效应的比率。之前的拟合结果主要取决于空间维度 n 和参数 \(\xi \in \mathbb {R}\) 的约束条件。这项工作解决了这个模型的强好拟性问题,首先是局部的,然后是全局的,对于小的初始数据,无论是在 \(L^p\)-\(L^2\)-setting for \(p > \frac{4}{4-n}\),还是在一般情况下,即对于 \(n = 2, 3\) 以及对 \(\xi \)没有任何限制。这种方法基于准线性方程的方法,以及相关线性化算子具有最大(L^p\)-(L^2\)规则性这一事实。后一个性质的证明依赖于扇形算子、舒尔互补和(\mathcal {J}\)对称性的技术。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
5.10
自引率
8.00%
发文量
98
审稿时长
4-8 weeks
期刊介绍: The Archive for Rational Mechanics and Analysis nourishes the discipline of mechanics as a deductive, mathematical science in the classical tradition and promotes analysis, particularly in the context of application. Its purpose is to give rapid and full publication to research of exceptional moment, depth and permanence.
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