Global existence of solutions of the Liquid Crystal flow for the Oseen-Frank model

TL;DR

This paper proves the global existence of solutions for the liquid crystal flow based on the Oseen-Frank model and extends these results to the Ericksen-Leslie system in two dimensions, advancing mathematical understanding of liquid crystal dynamics.

Contribution

It establishes the first global existence results for the liquid crystal flow under the Oseen-Frank model and extends these results to the Ericksen-Leslie system in two dimensions.

Findings

Global solutions exist for the liquid crystal flow in the Oseen-Frank model.

Global solutions are also established for the Ericksen-Leslie system in 2D.

The results contribute to the mathematical theory of liquid crystal dynamics.

Abstract

In the first part of this paper, we establish global existence of solutions of the liquid crystal (gradient) flow for the well-known Oseen-Frank model. The liquid crystal flow is a prototype of equations from the Ericksen-Leslie system in the hydrodynamic theory and generalizes the heat flow for harmonic maps into the 2-sphere. The Ericksen-Leslie system is a system of the Navier-Stokes equations coupled with the liquid crystal flow. In the second part of this paper, we also prove global existence of solutions of the Ericksen-Leslie system for a general Oseen-Frank model in $\Bbb R^2$.