Global well-posedness of the 2D nonhomogeneous incompressible nematic liquid crystal flows with vacuum

TL;DR

This paper proves the global existence and uniqueness of strong solutions for 2D nonhomogeneous incompressible nematic liquid crystal flows with vacuum, allowing large initial data and analyzing long-term behavior.

Contribution

It establishes the global well-posedness of the 2D flow model with vacuum and large initial data, under specific decay and geometric conditions.

Findings

Existence of unique global strong solutions

Initial data can include vacuum states and be arbitrarily large

Long-time behavior of solutions is characterized

Abstract

This paper concerns the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible nematic liquid crystal flows on the whole space $\mathbb{R}^{2}$ with vacuum as far field density. It is proved that the 2D nonhomogeneous incompressible nematic liquid crystal flows admits a unique global strong solution provided the initial data density and the gradient of orientation decay not too slow at infinity, and the initial orientation satisfies a geometric condition (see \eqref{eq1.3}). In particular, the initial data can be arbitrarily large and the initial density may contain vacuum states and even have compact support. As a byproduct, the large time behavior of the solution is also obtained.