Compressible hydrodynamic flow of nematic liquid crystals with vacuum

In this paper, we first consider the free boundary problem for a simplified version of Ericksen-Leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals in dimension one which connects continuously to vacuum. We obtain the existence of global weak solutions. Furthermore, we establish the life-span of smooth solutions to the compressible nematic liquid crystal model with the support of density growing sublinearly in time direction.