Global solvability of 3D non-isothermal incompressible nematic liquid crystal flows

We are concerned with the initial value problem of non-isothermal incompressible nematic liquid crystal flows in ${\mathbb{R}}^3$. Through some time-weighted a priori estimates, we prove the global existence of a strong solution provided that $(\Vert \sqrt{{\rho }_0}{u}_0{\Vert }_{L^2}^2+\Vert \mathrm{\nabla }{d}_0{\Vert }_{L^2}^2)(\Vert \mathrm{\nabla }{u}_0{\Vert }_{L^2}^2+\Vert {\mathrm{\nabla }}^2{d}_0{\Vert }_{L^2}^2)$ is reasonably small, which extends the corresponding Li's (Methods Appl. Anal. 2015 ^[4]) and Ding-Huang-Xia's (Filomat 2013 ^[2]) results to the whole space ${\mathbb{R}}^3$ and non-isothermal case. Furthermore, we also derive the algebraic decay estimates of the solution.