On static and hydrodynamic biaxial nematic liquid crystals

Abstract

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 $(n,m):\Omega \to {\mathbb{S}}^2\times {\mathbb{S}}^2$ 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.

arXiv comment37 pages

References in corpus (5)

• Recent developments of analysis for hydrodynamic flow of nematic liquid crystals
• Regularity and existence of global solutions to the Ericksen-Leslie system in ${\mathbb{R}}^2$
• Finite time singularity of the nematic liquid crystal flow in dimension three
• Finite time blow-up for the nematic liquid crystal flow in dimension two
• Decay estimates for Rivière's equation, with applications to regularity and compactness

Cited by in corpus (2)

• Weak compactness of simplified nematic liquid flows in 2D
• Frame hydrodynamics of biaxial nematics from molecular-theory-based tensor models