Local and global solutions on arcs for the Ericksen-Leslie problem in RN

The work deals with the Ericksen-Leslie system for nematic liquid crystals on the space R(N )with N >= 3. In our work, we suppose the initial condition v(0 )stayson 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 u is an element of L-infinity ((0, T); H-s(R-N)), del u is an element of L-2((0, T);H-s(R-N)), del v is an element of L-infinity((0, T);H-s(R-N)), del(2)v is an element of L-2((0, T);H-s(R-N)) for s > N/2 - 1, asking low regularity assumptions on u(0) and v(0)