Global Well-Posedness with Large Oscillations and Vacuum to the   Three-Dimensional Equations of Compressible Nematic Liquid Crystal Flows
  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. 
Jing Li, Zhonghai Xu, Jianwen Zhang
math.AP