Global Solutions of Nematic Liquid Crystal Flow in Dimension Two

Abstract

In this article we are concerned with a simplified Ericksen-Leslie system on $\mathbb{R}^2,$ whose bounded domain case was considered in [Lin et al., Arch. Ration. Mech. Anal. 197 (2010), 297–336]. With a study of its vorticity-stream formulation, we establish a global existence result of weak solutions when initial orientation has finite energy and initial vorticity function lies in $L^1 (\mathbb{R}^2).$