Well-posedness of frame hydrodynamics for biaxial nematic liquid crystals

TL;DR

This paper studies the mathematical well-posedness of a hydrodynamic model for biaxial nematic liquid crystals, establishing local and global solutions and blow-up criteria in two and three dimensions.

Contribution

It provides the first rigorous analysis of well-posedness and solution behavior for the frame hydrodynamics of biaxial nematic liquid crystals derived from molecular models.

Findings

Local well-posedness in 2D and 3D

Global existence of weak solutions in 2D

Blow-up criteria for smooth solutions

Abstract

We consider the hydrodynamics for the biaxial nematic phase characterized by a field of orthonormal frame, which can be derived from a molecular-theory-based tensor model. In dimension two and three, we establish the local well-posedness and the blow-up criterion for smooth solutions to the frame hydrodynamic model. Furthermore, we prove the global existence of weak solutions in $\mathbb{R}^2$ which are nonsmooth at finitely many singular times.