Global Existence and Large-Time Behavior of Strong Solutions to the Two-Dimensional Compressible Nematic Liquid Crystal Flows with Large Initial Data and Vacuum

Abstract

In this paper, we investigate two-dimensional compressible nematic liquid crystal flows with a planar director. We assume that the shear viscosity is a positive constant and that the bulk viscosity is given by $\lambda (\rho )={\rho }^{\beta }$ with $\beta >4/3$. In the periodic domain or in a bounded simply connected smooth domain with Navier-slip and Neumann boundary conditions, we establish the global existence and uniqueness of strong solutions without any restrictions on the size of the initial data. In particular, no geometric angle condition is imposed on the initial orientation field. Moreover, we derive a time-uniform upper bound for the density and establish exponential decay estimates for the strong solutions.