Local and global solutions on arcs for the Ericksen–Leslie problem in RN$\mathbb {R}^N$

Abstract

The work deals with the Ericksen–Leslie system for nematic liquid crystals on the space with . In our work, we suppose the initial condition stays on an arc connecting two fixed orthogonal vectors on the unit sphere. Thanks to this geometric assumption, we prove through energy a priori estimates the local existence and the global existence for small initial data of a solution for , asking low regularity assumptions on and .