The Existence Of The Solutions To The Incompressible Hydrodynamic Flow Of Biaxial Nematic Liquid Crystal

Liquid crystal is a kind of important materials.It has many excellent properties,such as high strength,high modulus,good flame retardancy,heat resistance,electro-optic effect,thermo-optical effect and anisotropy.So it has a very wide range of uses.For example,liquid crystal materials can be used to make display screens for electronic devices such as computers and televisions,to make lubricant and the shells of drug capsule,and even to research organisms and diagnosis of tumors.Therefore,the application of liquid crystal materials has been paid more and more attention,and the research of liquid crystal has become a hot spot.In this paper,we study the well-posedness for the incompressible hydrodynamic flow of biaxial nematic liquid crystal.First,when the space is two-dimensional or three-dimensional,we establish the iterative equations on the basis of the original equations,and prove the rationality of the iterative equations.Then prove that the sequence of the solutions to the iterative equations exist limit.Afterward,we take the limit of the subsequence of the iterative solutions and show that the limit we have taken is just the solution of the original equations.Finally,we explain the uniqueness of the solution.So the existence and uniqueness of the local solution for the non-negative initial density function and the existence and uniqueness of the local solution for the initial density function with positive lower bound are obtained respectively.In particular,when the space is two-dimensional,the global priori estimate of the solution is obtained by using the method of energy estimate,the Sobolev embedding theorem,the Schauder theory of parabolic equation and the theory of Navier-Stokes equations.Then combining with the existence and uniqueness of the local solution,we can get a conclusion that the existence and uniqueness of the global solution for the initial density function with positive lower bound can be obtained in the case of small initial values.