Weak solutions of non-isothermal nematic liquid crystal flow in dimension three

For any smooth domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$, we establish the existence of a global weak solution $(\mathbf{u},\mathbf{d},\theta )$ to the simplified, non-isothermal Ericksen-Leslie system modeling the hydrodynamic motion of nematic liquid crystals with variable temperature for any initial and boundary data $($\mathbf{u}$_0, $\mathbf{d}$0, \theta_0)\in$\mathbf{H}$\times H^1(\Omega, $\mathbb{S}$^2)\times L^1(\Omega)$,with$ $\mathbf{d}$0(\Omega)\subset$\mathbb{S}$+^2$(theupperhalfsphere)and$\displaystyle\inf\Omega \theta_0>0$.