关于静态和动态双轴向液晶
On static and hydrodynamic biaxial nematic liquid crystals
摘要：
在本文的第一部分中，我们将考虑具有三维$n\cdot m=0$的双轴向列$(n, m):Ω\to \mathbb S^2\times \mathbb S^2$的Oseen-Frank能量泛函$E(n, m)$的极小化构型，并建立它在$1$ -维Hausdorff测度零的闭集上是光滑的。在第二部分中，我们将考虑二维区域中双轴向列方程$(n, m)$的简化Ericksen-Leslie方程组，并建立了对于任何有限能量的初始和边界数据，存在唯一的整体弱解$(u, n, m)$，该解最多可光滑化200个奇异次。它们分别推广到Hardt-Kindelerherer-Lin \cite{HKL}和Lin-Lin-Wang \cite{LLW10}的简化的单轴液晶流体力学的相应结果。
英文摘要：
In the first part of this paper, we will consider minimizing configurations of the Oseen-Frank energy functional $E(n, m)$ for a biaxial nematics $(n, m):Ω\to \mathbb S^2\times \mathbb S^2$ with $n\cdot m=0$ in dimension three, and establish that it is smooth off a closed set of $1$-dimension Hausdorff measure zero. In the second part, we will consider a simplified Ericksen-Leslie system for biaxial nematics $(n, m)$ in a two dimensional domain and establish the existence of a unique global weak solution $(u, n, m)$ that is smooth off at most finitely many singular times for any initial and boundary data of finite energy. They extend to biaxial nematics of earlier results corresponding to minimizing uniaxial nematics by Hardt-Kindelerherer-Lin \cite{HKL} and a simplified hydrodynamics of uniaxial liquid crystal by Lin-Lin-Wang \cite{LLW10} respectively.
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