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

In this paper, we study the 2D incompressible nematic liquid crystal equations in smooth bounded domain, where the velocity u and macroscopic molecular orientation d admit the Dirich-let 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 allowing to vanish. Furthermore, this result extends the corresponding J. Li’s (Methods Appl. Anal. 2015 [11]) and X. Li’s (Discrete Contin. Dyn. Syst. 2017 [12]) results to the Neumann boundary condition, and removes the smallness conditions on the initial data. 2010MSC: 76A15; 35B65