Global Regularity to the liquid crystal flows of Q-tensor model

TL;DR

This paper proves global regularity for liquid crystal Q-tensor flows coupled with Navier-Stokes equations in bounded domains, using maximum principles and regularity criteria based on initial data norms.

Contribution

It extends previous results by establishing global strong solutions with small initial data and provides new regularity criteria based on initial data norms.

Findings

Existence of global strong solutions for small initial data in bounded domains.

Continuous dependence of solutions on initial data.

A regularity criterion based on the $ abla u$ norm and initial data norms in unbounded domains.

Abstract

In this paper we investigate a forced incompressible Navier-Stokes equation coupled with a parabolic type equation of Q-tensors in a domain $U\subset\R^3.$ In the case $U$ is bounded, we prove the existence of a global strong solution when the initial data are sufficiently small, improving a result in Xiao's paper [J. Differ. Equations 2017]. The key tool of the proof is a {maximum principle.} Then, we establish also a result of continuous dependence of solutions on the initial data. Finally, if $U=\R^3,$ based on a result of Du, Hu and Wang [Arch. Rational Mech. Anal. 2020], we give an interesting regularity criterium just via the $\dot{B}^{-1}_{\infty,\infty}$ norm of $u$ and the $L^\infty$ norm of the initial data $Q_0$.