Global existence for slightly compressible hydrodynamic flow of liquid crystals in two dimensions

In two dimensions, we study the compressible hydrodynamic flow of liquid crystals with periodic boundary conditions. As shown by Ding et al. （2013）, when the parameter λ→∞ oo, the solutions to the compressible liquid crystal system approximate that of the incompressible one. Furthermore, Ding et al. （2013） proved that the regular incompressible limit solution is global in time with small enough initial data. In this paper, we show that the solution to the compressible liquid crystal flow also exists for all time, provided that is sufficiently large and the initial data are almost incompressible.