The Vanish Limit Problem Of Liquid Crystal Flow Equations In Two Dimensions

The liquid crystal equations are proposed from the reach of liquid crystal dynamics.Liquid crystal is a material that has both crystal anisotropy and liquid flow,such as display.So the study of liquid crystal flow is very meaningful.But,liquid crystal equations are coupling between the Navier-Stokes equation and the transported flow of harmonic maps.Therefore,the study of liquid crystal equations have a lot of difficulty.The purpose of this paper is to research the limit behavior of the weak solutions of the liquid crystal equations in finite time as the resistance is reduced.Based on the purpose,this thesis is arranged as follows:We first introduce the background and the significance of liquid crystal equations.Then review the research history of the development of the liquid crystal equations and several important lemmas.Based on previous studies,we use the parabolic Morrey's decay lemma in spacetime and some important inequalities to show the regularity of the weak solutions of the liquid crystal equations under the bounded energy and the Gilbert damping coefficient is a constant.Using the Principle of Contraction Mapping and Schauder theorem to prove the existence of the local solutions of liquid crystal equations,and prove the existence of solutions of the limit equations.In L² space,we prove the weak solutions of the liquid crystal equations tend to the solution of the limit equations at the same initial boundary conditions,as the Gilbert damping coefficient approaches zero.At H₀¹,C^? spaces,discuss the conditions which the weak solutions of the liquid crystal equations tend to the solutions of the limit equations as the Gilbert's damping coefficient approaches zero.And obtain the system energy of the liquid crystal equations is less than or equal to the energy of the initial moment as resistance is trending to zero.