Local and global solutions on arcs for the Ericksen–Leslie problem in R^N

The work deals with the Ericksen–Leslie system for nematic liquid crystals on the space (Formula presented.) with (Formula presented.). In our work, we suppose the initial condition (Formula presented.) 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 (Formula presented.) (Formula presented.) for (Formula presented.), asking low regularity assumptions on (Formula presented.) and (Formula presented.).