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

Abstract

This paper considers a simplified three dimensional Ericksen-Leslie System for nematic liquid crystal flows in the unbounded domain $ \Omega: = \mathbb R^+\times \mathbb R^2 $ or the smooth bounded domain $ \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}(\Omega)\times H^{s}(\Omega) $, where the index $ s $ of the Sobolev space can be any large fixed integer, but $ s\neq+\infty $. Then we give an asymptotic expansions of a family of Sobolev regularity solutions for such system in $ \Omega $.

Keywords

• nematic liquid crystal flow
• global sobolev solution
• asymptotic expansion