Regularity and existence of global solutions to the Ericksen-Leslie system in $\mathbb R^2$

TL;DR

This paper proves regularity and global existence of weak solutions for the Ericksen-Leslie system in two dimensions, extending previous results to more general initial data and physical conditions.

Contribution

It establishes regularity and existence of global weak solutions for the full Ericksen-Leslie system in 2D under physical constraints, broadening prior simplified models.

Findings

Regularity theorem for weak solutions in 2D

Existence of global weak solutions for initial data in energy space

Solutions are smooth except at finitely many times

Abstract

In this paper, we first establish the regularity theorem for suitable weak solutions to the Ericksen-Leslie system in dimensions two. Building on such a regularity, we then establish the existence of a global weak solution to the Ericksen-Leslie system in $\mathbb R^2$ for any initial data in the energy space, under the physical constraint conditions on the Leslie coefficients ensuring the dissipation of energy of the system, which is smooth away from at most finitely many times. This extends earlier works by Lin,Lin, and Wang on a simplified nematic liquid crystal flow in dimensions two.