Existence Of Solutions To The Biaxial Nematic Liquid Crystals With Two Order Parameter Tensors

Liquid crystal is the fourth form of matter distinct from solid,liquid and gas.Liquid crystal can not only move freely like fluid,but also has anisotropy similar to that of crystal.As a kind of soft material,liquid crystal is widely used.In the development of liquid crystal displays,nematic liquid crystal is the fastest growing.According to the symmetry of liquid crystal,it can be divided into the uniaxial nematic liquid crystal and biaxial nematic liquid crystal.because biaxial nematic liquid crystal has fast response speed,and many other advantages,it can satisfy the demand of the rapid development of the display field,therefore,it is necessary to do some mathematical study on biaxial nematic liquid crystals.In this paper,we study the existence of classical solution and weak solution to the biaxial nematic liquid crystal with two order parameter tensors.To prove the existence and uniqueness of classical solution,first of all,the local existence and uniqueness of classical solutions can be obtained from the fixed point theorem,and then we prove the global existence of classical solution by a prior estimate,this part uses the maximum principle,interpolation inequality and Young inequality to complete the prior estimate.To prove the existence of the global weak solution,firstly,we expand the equation into component form,then we establish the approximation equation of the initial boundary value problem,then prove the existence of the global weak solution of the approximation equation by Galerkin method,then make some uniform estimation,and finally take the limit to obtain the existence of the global weak solution of the initial boundary value problem.this part using the Cauchy inequality,Young inequality,and interpolation to complete the prior estimate.