| Liquid crystal is a kind of typical and important complex fluid.Its unique anisotropic properties(such as phase transition,defect and dynamic behavior)provide a lot of mathematical problems with rich connotation and high challenge.The mathematical models of liquid crystal dynamics are Doi-Onsager molecular model,Landau-de Gennes tensor model and Ericksen-Leslie vector model.These three kinds of liquid crystal models are derived from different physical points of view.They have their own advantages and disadvantages: Doi-Onsager model is based on microscopic statistical mechanics,which considers the microscopic interaction between molecules,so it is more accurate,but usually involves high-dimensional problems,and it is difficult to solve.Landau-de Gennes model and Ericksen-Leslie model are based on macroscopical continuum mechanics.They are easy to apply and have good effect on the characterization of nematic phase,but the physical meaning of their parameters is not clear.Therefore,it is an important and basic problem to understand and establish the relationship between different level models in the study of liquid crystal mathematical theory.Qian-Sheng model is a kind of representative mathematical model to describe the dynamic behavior of liquid crystal in the framework of Landau-de Gennes tensor theory.In addition to describing the uniaxial nematic phase,the model can also describe more complex physical phenomena such as biaxial phase and line defect.In this paper,we mainly studies the relationship between the Qian-Sheng model ignoring inertia and the smooth solution of the Ericksen-Leslie model.For the de Gennes vector model describing the layered A-phase liquid crystal,we study the stability of the three-dimensional radial symmetric point defect structure.The specific research contents are as follows:Firstly,for the Qian-Sheng model with small elastic parameters,the solution is expanded by Hilbert expansion method,and a series of tensor equations are derived.With the help of critical point and the properties of linearization operator,the Ericksen-Leslie equation is derived from the tensor equation(i.e.zero order system)in the equilibrium state.For the first-order and second-order systems obtained by Hilbert expansion,two projection operators are used to transform them into closed linear equations.The existence of Hilbert expansion is proved by standard energy method.A ”good term” is introduced to simplify the equation system satisfied by the remainder.The characteristic of ”good term” is that when the elastic small parameter tends to zero(also known as the uniaxial limit),it can be controlled by a given energy functional.With the help of the key estimation of the singular term and the cancellation relation,the uniform energy estimation of the remainder in the Hilbert expansion is proved.Therefore,it is proved mathematically that the smooth solution of non-inertial Qian-Sheng model converges to the smooth solution of Ericksen-Leslie model in the sense of uniaxial limit.In addition to the rich phase structure,liquid crystal also has a very important physical phenomenon is the existence of defects.Intuitively,the defect is the place where the local molecules are arranged discontinuously.In the equilibrium state,the position and configuration of the defect structure are determined by the minimal free energy solution and the boundary constraint.By minimizing the free energy functional of smectic crystal A-phase,we can obtain the equilibrium solution with its region confined in three-dimensional sphere,and then prove the stability of its point defect configuration with radial symmetry by the second-order variational method. |
