Global strong solutions for non-isothermal compressible nematic liquid crystal flows under a scaling-invariant smallness condition
Lin Xu, Xin Zhong
TL;DR
This work studies the 3D non-isothermal compressible nematic liquid crystal system with vacuum in the far-field and proves global existence and uniqueness of strong solutions under a scaling-invariant smallness condition. A novel scaling-invariant quantity is identified, and a refined set of a priori estimates is developed to close a bootstrap argument without imposing the previously required viscosity constraint $2\mu>\lambda$. The analysis relies on new density bounds in $L^{\infty}_tL^3_x$, direct handling of the temperature equation to bound $\nabla\theta$, and careful control of the director field $d$ interactions. The result advances the theory by removing the viscosity restriction and clarifying the role of scaling-invariant quantities in governing global well-posedness for this coupled fluid-director system.
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: $$ \big(1+\barρ+\tfrac{1}{\barρ}\big) \big[\|ρ_{0}\|_{L^{3}}+(\barρ^{2}+\barρ)\big(\|\sqrt{ρ_{0}}u_{0}\|_{L^{2}}^{2}+\|\nabla d_{0}\|_{L^{2}}^{2}\big)\big] \big[\|\nabla u_{0}\|_{L^{2}}^{2}+(\barρ+1)\|\sqrt{ρ_{0}}θ_{0}\|_{L^{2}}^{2} +\|\nabla^{2} d_{0}\|_{L^{2}}^{2}+\|\nabla d_{0}\|_{L^{4}}^{4}\big]. $$ 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).