Global Low-Energy Weak Solution and Large-Time Behavior for the Compressible Flow of Liquid Crystals

We consider the weak solution of the simplified Ericksen-Leslie system modeling compressible nematic liquid crystal flows in ${\mathbb{R}}^3$. When the initial data is small in $L^2$ and initial density is positive and essentially bounded, we first prove the existence of a global weak solution in $\mathbb R^3$. The large-time behavior of a global weak solution is also established.