On The Regularity Criteria Of Solutions To Liquid Crystal Flow In Three Dimensions

The liquid crystal is a kind of widely used material materials,which is an important re-search object of physics,chemistry and materials science.At present,the research related to liquid crystal mathematical theory is still in its preliminary stage.Although the mathematical models of liquid crystals are complex,their mathematical structures are beautiful,and they can be described by molecular theory,vector theory,tensor theory and other angles.These models provide rich and meaningful problems for the study of partial differential equations.In partic-ular,liquid crystal dynamics models are important research objects in complex hydrodynamic partial differential equations.This paper studies the Ericksen-Leslie model described by vector theory,which is a coupling of the order parameter evolution equation and the Navier-Stokes equation.We mainly studies the regularity of the solution to this system.The structure of this article is as follows:In Chapter 1,we outline the background of problems and research status at home and abroad.In Chapter 2,we consider the regularity criterion problem of the Ginzburg-Landau approxi-mation liquid crystal model in R3.Given that the initial values u0,d0 satisfy certain assuPptions,we first use the energy method,Sobolev inequalities and analysis tools to estimate energy equa-tions which are about horizontal gradient of the velocity field Vhu and the second derivative components of orientation field ?h?d.We derive the energy equations of ?u and ?d.And then combining the assumptions of the velocity field component u3 and the horizontal gradient of orientation field ?hd,we prove that solution to the three-dimensional liquid crystal equations can be extended beyond T.In Chapter 3,we consider the regularity problem of the liquid crystal model in R3under the assumption of the pressure field gradient component(?)x3 P and the molecular direction field gradient components ?hd,(?)x3d.First of all,the energy equation of the velocity field component u3 is derived by the energy method,and combining the given hypothesis of pressure field gra-dient component(?)X3P and the horizontal gradient components of the molecular direction field?hd,(?)x3d,we estimate the velocity field component u3.Then combining the results in Chapter 2,we prove that solution to the three-dimensional liquid crystal equations can be extended beyond T.In Chapter 4,we continue to discuss the regularity criterion problem of liquid crystal model in R3.Similar to Chapter 2,we first use the energy method to derive the energy equation of the molecular direction field gradient component(?)x3d,,combining the assumption of the given ve-locity field horizontal components uh,such that we obtain a certain norm estimate of the molec-ular direction field gradient component(?)3d.Then by the energy method,we derive the energy equations of ?u and ?d,and combining the above norm estimate of the gradient component of the molecular direction field(?)x3d,such that we prove that solution to the three-dimensional liquid crystal equations can be extended beyond T.In addition,combining the energy equation of the velocity field horizontal gradient ?hu,under the assumptions of pressure field horizontal gradient components ?hP and the molecular direction field gradient ?d,estimating them step by step,we prove that solution to the three-dimensional liquid crystal equations can be extended beyond T.In Chapter 5,we made a summary and introduced the future work.