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

Abstract

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: $$(1+\bar{\ddot{I}}+\frac{1}{\bar{\ddot{I}}})[\Vert \ddot{I}{}_0{\Vert }_{L^3}+(\bar{\ddot{I}}{}^2+\bar{\ddot{I}})(\Vert \sqrt{\ddot{I}{}_0}{u}_0{\Vert }_{L^2}^2+\Vert \nabla d_0{\Vert }_{L^2}^2\left)\right][\Vert \nabla u_0{\Vert }_{L^2}^2+(\bar{\ddot{I}}+1)\Vert \sqrt{\ddot{I}{}_0}\hat{I}{¸}_0{\Vert }_{L^2}^2+\Vert {\nabla }^2{d}_0{\Vert }_{L^2}^2+\Vert \nabla d_0{\Vert }_{L^4}^4].$$ 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).

20 pages