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: $$(1+\bar{\rho }+\frac{1}{\bar{\rho }})[\mathrm{\parallel }{\rho }_0{\mathrm{\parallel }}_{L^3}+({\bar{\rho }}^2+\bar{\rho })(\mathrm{\parallel }\sqrt{{\rho }_0}{u}_0{\mathrm{\parallel }}_{L^2}^2+\mathrm{\parallel }\mathrm{\nabla }{d}_0{\mathrm{\parallel }}_{L^2}^2\left)\right][\mathrm{\parallel }\mathrm{\nabla }{u}_0{\mathrm{\parallel }}_{L^2}^2+(\bar{\rho }+1)\mathrm{\parallel }\sqrt{{\rho }_0}{\theta }_0{\mathrm{\parallel }}_{L^2}^2+\mathrm{\parallel }{\mathrm{\nabla }}^2{d}_0{\mathrm{\parallel }}_{L^2}^2+\mathrm{\parallel }\mathrm{\nabla }{d}_0{\mathrm{\parallel }}_{L^4}^4]\mathrm{.}$$ 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).