  We consider the strong and weak solutions to the Cauchy problem of the
inhomogeneous incompressible nematic liquid crystal equations in two
dimensions. We first establish the local existence and uniqueness of strong
solutions by using the standard domain expanding method, and then extend such
local strong solution to be a global one, provided the initial density is away
from vacuum and the initial direction field satisfies some geometric structure.
The size of the initial data can be large. Based on such global existence
results of strong solutions, by using compactness argument, we obtain the
global existence of weak solutions with nonnegative initial density.
