  For any bounded smooth domain $\Omega\subset\mathbb R^3$, we establish the
global existence of a weak solution $u:\Omega\times (0,+\infty)\to\mathbb
R^3\times\mathbb S^2$ of the initial-boundary value (or the Cauchy) problem of
the simplified Ericksen-Leslie system (1.1) modeling the hydrodynamic flow of
nematic liquid crystals for any initial and boundary (or Cauchy) data $(u_0.
d_0)\in {\bf H}\times H^1(\Omega,\mathbb S^2$), with $d_0(\Omega)\subset\mathbb
S^2_+$
  (the upper hemisphere). Furthermore, ($u,d$) satisfies the global energy
inequality (1.4).
