Dynamics of Nematic Liquid Crystal Flows: the Quasilinear Approach

TL;DR

This paper develops a comprehensive dynamic theory for nematic liquid crystal flows using a quasilinear parabolic approach, proving existence, uniqueness, and exponential convergence of solutions in an $L_p-L_q$ framework.

Contribution

It introduces a novel quasilinear parabolic framework for analyzing nematic liquid crystal flow equations, establishing well-posedness and long-term behavior results.

Findings

Existence of unique local strong solutions.

Global solutions exist near equilibrium or when bounded.

Solutions converge exponentially to equilibrium.

Abstract

Consider the (simplified) Leslie-Erickson model for the flow of nematic liquid crystals in a bounded domain $\Omega \subset \mathbb{R}^n$ for n > 1$. This article develops a complete dynamic theory for these equations, analyzing the system as a quasilinear parabolic evolution equation in an $L_p-L_q$-setting. First, the existence of a unique local strong solution is proved. This solution extends to a global strong solution, provided the initial data are close to an equilibrium or the solution is eventually bounded in the natural norm of the underlying state space. In this case the solution converges exponentially to an equilibrium. Moreover, the solution is shown to be real analytic, jointly in time and space.