Global Strong Solutions to Incompressible Nematic Liquid Crystal Flow

TL;DR

This paper proves the global existence and uniqueness of strong solutions for the inhomogeneous incompressible nematic liquid crystal equations in bounded domains, allowing for initial vacuum and small initial energy in two and three dimensions.

Contribution

It establishes the first global strong solution results for these equations with small initial energy and vacuum, using novel a priori estimates.

Findings

Global strong solutions exist and are unique in 2D and 3D under small initial energy.

The solutions can be extended globally in time from local solutions.

The approach relies on establishing time-independent a priori estimates.

Abstract

In this paper, we consider the Dirichlet problem of inhomogeneous incompressible nematic liquid crystal equations in bounded smooth domains of two or three dimensions. We prove the global existence and uniqueness of strong solutions with initial data being of small norm but allowed to have vacuum. More precisely, for two dimensional case, we only require that the basic energy $|\sqrt{\rho_0}u_0|_{L^2}^2+|\nabla d_0|_{L^2}^2$ is small, while for three dimensional case, we ask for the smallness of the production of the basic energy and the quantity $|\nabla u_0|_{L^2}^2+|\nabla^2d_0|_{L^2}^2$. Our efforts mainly center on the establishment of the time independent a priori estimate on local strong solutions. Taking advantage of such a priori estimate, we extend the local strong solution to the whole time, obtaining the global strong solution.