Global solution to the nematic liquid crystal flows with heat effect

TL;DR

This paper establishes the existence of global solutions for temperature-dependent nematic liquid crystal flows in bounded domains, extending previous results to include heat effects and variable viscosity.

Contribution

It generalizes prior work by incorporating heat effects into the analysis of nematic liquid crystal flows, proving global existence under small initial perturbations.

Findings

Proved local existence of strong solutions.

Extended to global solutions with small initial data.

Generalized previous results to non-isothermal cases.

Abstract

The temperature-dependent incompressible nematic liquid crystal flows in a bounded domain $\Omega\subset\mathbb{R}^N$ ($N=2, 3$) are studied in this paper. Following Danchin's method in [J. Math. Fluid Mech., 2006], we use a localization argument to recover the maximal regularity of Stokes equation with variable viscosity, by which we first prove the local existence of strong solution, then extend it to a global one provided that the initial data is a sufficiently small perturbation around the trivial equilibrium state. This paper also generalizes Hu-Wang's result in [Commun. Math. Phys., 2010] to the non-isothermal case.