On The Incompressible Nematic Liquid Crystal Flow In Weak-L~n Spaces

In this paper,the Cauchy problem of incompressible nematic liquid crystal flow in Rn is considered.ut + u · ?u-?u +?p =-? ·(?d??d),in Rn ×(0,?),? · u = 0,in Rn ×(0,?),dt + u · ? = Ad + |?d|2d,in Rn x(0,?),with the following initial data condition u(x,0)= u0(x),d(x,0)=d0(x),in Rn,(0-1)By applying the time-space estimates for the heat equation and stokes equation,we prove that when initial data belongs to the weak-Ln spaces,there exists a global mild solutions when the initial data is small enough.Moreover,the stability of those mild solutions in relation to a perturbation of the initial data is considered.We are interested in the asymptotic stability of mild solutions as the time goes to infinity.This article is divided into four chapters.The first chapter is the introduction,which introduces the significance,background and the research results of the incompressible liquid crystal flow.The second chapter is the basic knowledge.The third chapter is to prove the existence of global solutions.The fourth chapter is to prove the asymptotic stability of mild solutions.