Existence of Sobolev regular solutions for the incompressible flow of liquid crystals in three dimensions

This paper considers a simplified three dimensional Ericksen-Leslie System for nematic liquid crystal flows in the unbounded domain $\mathrm{\Omega }:={\mathbb{R}}^{+}\times {\mathbb{R}}^2$ or the smooth bounded domain $\mathrm{\Omega }$. The hydrodynamic system consists of the Navier-Stokes type equations for the fluid velocity coupled with a convective Ginzburg-Landau type equation for the averaged molecular orientation. We first establish the global existence of Sobolev regular solution with finite energies in Sobolev space $H^s(\mathrm{\Omega })\times H^s(\mathrm{\Omega })$, where the index $s$ of the Sobolev space can be any large fixed integer, but $s\ne +\mathrm{\infty }$. Then we give an asymptotic expansions of a family of Sobolev regularity solutions for such system in $\mathrm{\Omega }$.