Global existence of strong solutions with large oscillations and vacuum to the compressible nematic liquid crystal flows in 3D bounded domains

Abstract

We investigate compressible nematic liquid crystal flows in three-dimensional (3D) bounded domains with slip boundary condition for velocity and Neumann boundary condition for orientation field. By applying piecewise-estimate method and delicate analysis based on the effective viscous flux and vorticity, we derive the global existence and uniqueness of strong solutions provided that the initial total energy is suitably small. Our result is an extension of the works of Huang-Wang-Wen (J. Differential Equations 252: 2222-2265, 2012) and Li-Xu-Zhang (J. Math. Fluid Mech. 20: 2105-2145, 2018), where the local strong solutions in three dimensions and the global strong solutions for 3D Cauchy problem were established, respectively. Moreover, it also shows that blow up mechanism for local strong solutions obtained by Huang-Wang-Wen (Arch. Ration. Mech. Anal. 204: 285-311, 2012) cannot occur if the initial total energy is sufficiently small.