  The Cauchy problem for the compressible flow of nematic liquid crystals in
the framework of critical spaces is considered. We first establish the
existence and uniqueness of global solutions provided that the initial data are
close to some equilibrium states. This result improves the work by Hu and Wu
[SIAM J. Math. Anal., 45 (2013), pp. 2678-2699] through relaxing the regularity
requirement of the initial data in terms of the director field. We then
consider the incompressible limit problem for ill prepared initial data. We
prove that as the Mach number tends to zero, the global solution to the
compressible flow of liquid crystals converges to the solution to the
corresponding incompressible model in some function spaces. Moreover, the
accurate converge rates are obtained.

Analysis of PDEs