Well-posedness of the Ericksen-Leslie system

TL;DR

This paper proves local and global well-posedness of the Ericksen-Leslie system for liquid crystals, introducing a new formulation that ensures energy dissipation without relying on Ginzburg-Landau approximation.

Contribution

It establishes well-posedness results for the Ericksen-Leslie system using a novel formulation that guarantees energy dissipation, improving upon previous methods.

Findings

Local well-posedness proved

Global well-posedness for small initial data

New formulation ensures energy dissipation

Abstract

In this paper, we prove the local well-posedness of the Ericksen-Leslie system, and the global well-posednss for small initial data under the physical constrain condition on the Leslie coefficients, which ensures that the energy of the system is dissipated. Instead of the Ginzburg-Landau approximation, we construct an approximate system with the dissipated energy based on a new formulation of the system.