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

Abstract

For any smooth domain $\Omega\subset \mathbb{R}^3$, 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$ (the upper half sphere) and $\displaystyle\inf_\Omega \theta_0>0$.