  In this paper, we study the Cauchy problem of the Poiseuille flow of full
Ericksen-Leslie model for nematic liquid crystals. The model is a coupled
system of a parabolic equation for the velocity and a quasilinear wave equation
for the director. For a particular choice of several physical parameter values,
we construct solutions with smooth initial data and finite energy that produce,
in finite time, cusp singularities - blowups of gradients. The formation of
cusp singularity is due to local interactions of wave-like characteristics of
solutions, which is different from the mechanism of finite time singularity
formations for the parabolic Ericksen-Leslie system. The finite time
singularity formation for the physical model might raise some concerns for
purposes of applications. This is, however, resolved satisfactorily; more
precisely, we are able to establish the global existence of weak solutions that
are H\"older continuous and have bounded energy. One major contribution of this
paper is our identification of the effect of the flux density of the velocity
on the director and the reveal of a singularity cancellation - the flux density
remains uniformly bounded while its two components approach infinity at
formations of cusp singularities.
