Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density

TL;DR

This paper proves the global existence and uniqueness of strong solutions for 2D nonhomogeneous incompressible liquid crystal flows, allowing vacuum states and large initial velocities, under certain smallness and geometric conditions.

Contribution

It establishes the first global well-posedness results for this model with vacuum and large data, introducing a Serrin-type criterion based on the gradient of the director field.

Findings

Global strong solutions exist under small initial data conditions.

Vacuum states in initial density are permitted.

Large initial velocities are allowed with geometric angle conditions.

Abstract

In this paper, the authors first establish the global well-posedness of strong solutions of the simplified Ericksen-Leslie model for nonhomogeneous incompressible nematic liquid crystal flows in two dimensions if the initial data satisfies some smallness condition. It is worth pointing out that the initial density is allowed to contain vacuum states and the initial velocity can be arbitrarily large. We also present a Serrin's type criterion, depending only on $\nabla d$, for the breakdown of local strong solutions. As a byproduct, the global strong solutions with large initial data are obtained, provided the macroscopic molecular orientation of the liquid crystal materials satisfies a natural geometric angle condition (cf. [19])