Research On Weak Solutions Of Incompressible Biaxial Nematic Liquid Crystal Dynamical System With Double Tensor

Liquid crystal is a substance between liquid and crystal.Although liquid crystal loses the rigidity of solid material,it has the fluidity of liquid.Liquid crystal molecules can be oriented,so liquid crystal has anisotropy similar to crystalline material.Liquid crystal materials also have good flame retardancy and electro-optical effects,so liquid crystals are widely used as materials.Nematic liquid crystals are divided into uniaxial nematic and biaxial nematic liquid crystals.Biaxial nematic liquid crystals are also used widely.Therefore,it is necessary to study the mathematical problems in biaxial nematic liquid crystal.In this paper,we study the existences of weak solutions to the incompressible biaxial nematic liquid crystal dynamics system with double tensors,we obtain the global existences of weak solutions to this system with time-dependent or independent boundary condition.For the case of time-independent boundary condition,we first obtain the energy equation,and then prove the symmetry and zero trace retention of QB system.Then,we construct a proximate solution by Galerkin method.The convergence of the approximation solution sequence is obtained by using a prior estimation.Finally,we get the global existence of the weak solution with time-independent boundary condition.For the case of time-dependent boundary condition,the elliptic problem is used to improve the boundary value function.Then the boundary value condition problem can be transformed into the time-independent boundary value problem.We mainly derive a priori estimation of the system for this case.Finally,we obtain the weak solution by a similar method as the case of time-independent boundary condition.