| We know that all matter is composed of elementary particles. For example, each element of nature is composed of molecules, and molecule is composed of atoms, and so on. In many physical process, a state is the condition of a physical system with regard to phase, form, composition, or structure. Solid materials are formed from densely-packed atoms, with intense interaction forces between them. Depending on the material involved and the conditions in which it was formed, the atoms may be arranged in a regular, ge-ometric pattern (crystalline solids, whose defining characteristic is the periodicity of its atoms) or irregularly (an amorphous solid such as common window glass). Liquid crys-tals (LCs) are a state of matter that has properties between those of a conventional liquid and those of a solid crystal.This thesis is devoted to study the properties of Liquid crystals. Liquid crystals are partially ordered systems without a rigid, long-ranged structure. They are intermediate in symmetry and structure between the solid crystalline state and the amorphous liquid state. Due to their dual nature-anisotropic physical properties of solids and rheological behavior of liquids-and easy response to externally applied electric, magnetic, optical and surface fields, liquid crystals are of greatest potential for scientific and technological applications.In 1888, Austrian botanical physiologist Friedrich Reinitzer[1], working at the Charles University in Prague, examined the physico-chemical properties of various derivatives of cholesterol, which are now known as cholesteric liquid crystals. Previously, other re-searchers had observed distinct color effects when cooling cholesterol derivatives just above the freezing point, but had not associated it with a new phenomenon. Reinitzer perceived that color changes in a derivative cholesteryl benzoate were not the most pe-culiar feature. He found that cholesteryl benzoate does not melt in the same manner as other compounds, but has two melting points. At 145.5℃(293.9℉) it melts into a cloudy liquid, and at 178.5℃(353.3℉) it melts again and the cloudy liquid becomes clear. By that time, Reinitzer had discovered and described three important features of cholesteric liquid crystals (the name coined by Georges Friedel in 1922):the existence of two melting points, the reflection of circularly polarized light, and the ability to rotate the polarization direction of light. However, liquid crystals were not popular among scientists and the material remained a pure scientific curiosity for about 80 years [2]. Liquid crystals find wide use in liquid crystal displays (LCD), which rely on the optical properties of cer-tain liquid crystalline substances in the presence or absence of an electric field. Some liq-uid crystal materials change color when stretched or stressed. Thus, liquid crystal sheets are often used in industry to look for hot spots, map heat flow, measure stress distribution patterns, and so on. Liquid crystal in fluid form is used to detect electrically generated hot spots for failure analysis in the semiconductor industry [3] [4]. Liquid crystal memory units with extensive capacity were used in Space Shuttle navigation equipment [4].Liquid crystal equations arise not only from many fields of mathematics, but also from other branches of science such as physics, mechanics and material science. The complexity and challenges in the theoretical study of liquid crystal equations have at-tracted a lot of interests from many mathematicians for a long time.The present thesis is devoted to the study of the existence result of the liquid crystal system, the boundedness of the energy function of the system and the asymptotic behavior which based on the bounded energy of such finite energy weak solution as time tends to infinity. All the results obtained in this thesis have never been found in the previous literature.The thesis, containing 3 chapters, is meant for the recent result we have obtained. Chapter 1 is a preliminary chapter in which we not only recall the history in the litera-ture, but also illustrate the main idea of the proof of the existence of finite energy weak solutions. We discuss the new features and associated mathematical difficulties of the problems under consideration. Some basic materials and frequently used inequalities are also presented. Chapter 2 is concerned with the liquid crystal system amended by Ginzburg-Landau function with Dirichlet boundary conditions. Overcome the mathematical difficulties due to the nonlinear pressure p of density p, we obtain the existence of the global weak solution.Chapter 3 is concerned with the boundedness of the energy for the liquid crystals system and the asymptotic behavior of the finite energy weak solution. We show the convergence of the solution to a steady state as time tends to infinity.We briefly point out the new features, mathematical difficulties of the problems con-sidered in this thesis and our main contributions.First, in chapter 2 we consider the model of certain tensor, and use Feado-Galerkin approximation to get an approximate solution.Second, we do a prior estimate with the pressure function to get the convergent subsequence.
|
PDF Full Text Request