Global strong solutions for non-isothermal compressible nematic liquid crystal flows under a scaling-invariant smallness condition

We study the three-dimensional Cauchy problem for a non-isothermal compressible nematic liquid crystal system with far-field vacuum. By deriving refined energy estimates and exploiting the coupled structure of the equations, we establish the global existence and uniqueness of strong solutions, provided that the following scaling-invariant quantity is sufficiently small: $$\big(1+\bar{\rho}+\tfrac{1}{\bar{\rho}}\big) \big[\|\rho_{0}\|_{L^{3}}+(\bar{\rho}^{2}+\bar{\rho})\big(\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}}^{2}+\|\nabla d_{0}\|_{L^{2}}^{2}\big)\big] \big[\|\nabla u_{0}\|_{L^{2}}^{2}+(\bar{\rho}+1)\|\sqrt{\rho_{0}}\theta_{0}\|_{L^{2}}^{2} +\|\nabla^{2} d_{0}\|_{L^{2}}^{2}+\|\nabla d_{0}\|_{L^{4}}^{4}\big].$$ In particular, our result identifies a new scaling-invariant quantity and does not impose additional restrictions on the viscosity coefficients, which improves previous work (Commun. Math. Sci. 21 (2023), 1455--1486).