Incompressible limit of the Ericksen-Leslie hyperbolic liquid crystal model in compressible flow

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.