  Ericksen and Leslie proposed a hydrodynamic model for liquid crystals in the
format of conservation laws in the 1960s. Their original model includes
inertial and compressibility effects, which makes the model a coupled
parabolic-hyperbolic system. In this paper we build up the connection between
the compressible and incompressible parabolic-hyperbolic liquid crystal model
in the framework of classical solutions.
  We first derive the scaled Ericksen-Leslie system with dimensionless numbers,
including Mach, Reynolds, and Ericksen numbers. In particular, we introduce the
so-called inertial constant $\chi$ which characterizes the inertial effect of
the liquid crystal molecular. Next, we establish the energy estimates uniform
in the Mach number $\epsilon$ for both the compressible system and its
time-derivative system with small data. Then, we pass to the limit $\epsilon
\rightarrow 0$ in the compressible system, so that we establish the global
classical solution of the incompressible system by the compactness arguments.
Moreover, we also obtain the convergence rates associated with $L^2$-norm in
the case of well-prepared initial data. This is the first result on the
incompressible limit of the compressible parabolic-hyperbolic liquid crystal
model and confirms the relations of different parabolic-hyperbolic liquid
crystal model rigorously.
