Global solution to the nematic liquid crystal flows with heat effect

The temperature-dependent incompressible nematic liquid crystal flows in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$ ($N=2,3$) are studied in this paper. Following Danchin’s method in [J. Math. Fluid Mech., 2006], we use a localization argument to recover the maximal regularity of Stokes equation with variable viscosity, by which we first prove the local existence of strong solution, then extend it to a global one provided that the initial data is a sufficiently small perturbation around the trivial equilibrium state. This paper also generalizes Hu-Wang’s result in [Commun. Math. Phys., 2010] to the non-isothermal case.