Global Well-Posedness with Large Oscillations and Vacuum to the Three-Dimensional Equations of Compressible Nematic Liquid Crystal Flows

TL;DR

This paper proves the global existence of classical solutions for 3D compressible nematic liquid crystal flows with large oscillations and vacuum, allowing initial densities with compact support and analyzing long-term behavior.

Contribution

It establishes the first global well-posedness result for these flows with large oscillations and vacuum, even with initial densities of compact support.

Findings

Global classical solutions exist under large oscillations and vacuum conditions.

Initial vacuum regions can have arbitrarily large measure or compact support.

Long-time behavior of solutions is characterized.

Abstract

This paper is concerned with the three-dimensional equations of a simplified hydrodynamic flow modeling the motion of compressible, nematic liquid crystal materials. The authors establish the global existence of classical solution to the Cauchy problem with smooth initial data which are of small energy but possibly large oscillations with constant state as far-field condition which could be either vacuum or non-vacuum. The initial density is allowed to vanish and the spatial measure of the set of vacuum can be arbitrarily large, in particular, the initial density can even have compact support. As a byproduct, the large-time behavior of the solution is also studied.