Global small solutions to heat conductive compressible nematic liquid crystal system: smallness on a scaling invariant quantity

In this paper, we consider the Cauchy problem to the three dimensional heat conducting compressible nematic liquid crystal system in the presence of vacuum and with vacuum far fields. Global well-posedness of strong solutions is established under the condition that the scaling invariant quantity $ (\|\rho_0\|_\infty+1)\big[\|\rho_0\|_3+(\|\rho_0\|_\infty+1)^2(\|\sqrt{\rho_0}u_0\|_2^2+ \|\nabla d_0\|_2^2)\big] \big[\|\nabla u_0\|_2^2+(\|\rho_0\|_\infty+1)(\|\sqrt{\rho_0}E_0\|_2^2 + \|\nabla^2 d_0\|_2^2)\big]$ is sufficiently small with the smallness depending only on the parameters appeared in the system.