Global strong solution to the two-dimensional density-dependent nematic liquid crystal flows with vacuum

TL;DR

This paper proves the existence and uniqueness of global strong solutions for 2D density-dependent nematic liquid crystal flows with vacuum, under certain decay and smallness conditions on initial data, and analyzes their long-term behavior.

Contribution

It establishes the first global strong solution results for 2D nonhomogeneous nematic liquid crystal flows with vacuum and describes their asymptotic behavior.

Findings

Existence of unique global strong solutions under decay and smallness conditions.

Initial vacuum states are allowed, including compactly supported densities.

Long-term behavior of solutions is characterized.

Abstract

We are concerned with 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 basic energy $\|\sqrt{\rho_0}\mathbf{u}_0\|_{L^2}^2+\|\nabla\mathbf{d}_0\|_{L^2}^2$ is small. In particular, the initial density may contain vacuum states and even have compact support. Moreover, the large time behavior of the solution is also obtained.