Strong Solution to the Density-dependent Incompressible Nematic Liquid Crystal Flows

TL;DR

This paper proves local existence, uniqueness, and blowup criteria for density-dependent incompressible nematic liquid crystal flows in 2D and 3D, allowing initial densities that are not strictly positive.

Contribution

It establishes new blowup criteria based solely on the density gradient under certain initial geometric conditions, extending previous results.

Findings

Proved local existence and uniqueness of solutions.

Established blowup criteria for regularity in 2D and 3D.

Developed a density-gradient-based blowup criterion.

Abstract

In this paper, we investigate the density-dependent incompressible nematic liquid crystal flows in $n(n=2$ or $3)$ dimensional bounded domain. More precisely, we obtain the local existence and uniqueness of the solutions when the viscosity coefficient of fluid depends on density. Moreover, we establish blowup criterions for the regularity of the strong solutions in dimension two and three respectively. In particular, we build a blowup criterion just in terms of the gradient of density if the initial direction field satisfies some geometric configuration. For these results, the initial density needs not be strictly positive.