GLOBAL REGULARITY TO THE 2D INHOMOGENEOUS LIQUID CRYSTAL FLOWS WITH LARGE INITIAL DATA AND VACUUM

Abstract

We study the 2D incompressible nematic liquid crystal equations in a smooth bounded domain, where the velocity $u$ and macroscopic molecular orientation $d$ admit the Dirichlet and Neumann boundary condition, respectively. Under a geometric condition for the initial orientation field, we establish the global existence of strong solutions with large initial data. In particular, the initial density can be allowed to vanish. Furthermore, this result extends the corresponding results of J. Li (2015) and X. Li (2017) to the Neumann boundary condition and removes the smallness conditions on the initial data.