We investigate the three dimensional generalEricksen-Leslie (EL) system with Ginzburg-Landau
type approximation modeling nematic liquid crystal flows.First, we prove existence of global-in-time weak solutions under physically meaningful boundary conditions and suitable assumptions on the Leslie coefficients, which ensures that the total energy of the EL system is dissipated.
Moreover, for the EL system with periodic boundary conditions,we prove the local well-posedness of classical solutions under the so-called Parodi's relation and establish a blow-up criterion
in terms of the temporal integral of both the maximum norm of the curl of the velocity field and the maximum norm of the gradient of the liquid crystal director field.