Global existence of weak solutions of the nematic liquid crystal flow in dimensions three

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).