In this paper, we study the initial-boundary value problem for the Poiseuille
flow of hyperbolic-parabolic Ericksen-Leslie model of nematic liquid crystals
in one space dimension. Due to the quasilinearity, the solution of this model
in general forms cusp singularity. We prove the global existence of H\"older
continuous solution, which may include cusp singularity, for initial-boundary
value problems with different types of boundary conditions.Comment: 36 pages, 3 figure