Small data global regularity and scattering for 3D Ericksen-Leslie compressible hyperbolic liquid crystal model

We study the Ericksen-Leslie hyperbolic system for compressible liquid crystal model in three spatial dimensions. Global regularity and scattering for small and smooth initial data near equilibrium are proved for the case that the system is a nonlinear coupling of compressible Navier-Stokes equations with wave map to S². The main strategy relies on an interplay between the control of high order energies and decay estimates, which is based on the idea inspired by the method of space-time resonances. Unlike the incompressible model, the different behaviors of the decay properties of the density and velocity field for compressible fluids at different frequencies play a key role, which is a particular feature of compressible model.