Some Research On The Equations Describing The Hydrodynamics Of Nematic Liquid Crystals

The present Ph.D. dissertation is concerned with the well-posedness of the Ericksen-Lesile model, which can well describe the motion of nematic liquid crystals in experiments. In mathematics, the E-L model couples the Navier-Stokes equations with the harmonic heat flow equations. According to the kinds of liquid crystals, this model can be divided into the incompressible model and the compressible model. Both cases will be considered in this thesis. From the view of mathematics, there are two types of E-L model. One is Ginzburg-Landau (G-L) model, using Ginzburg-Landau penalization to approximate the unite director of liquid crystals. The other is non-Ginzburg-Landau (non-G-L) model, which is characterized by the angular momentum equation with the second order growth gradient term|▽d|2d, similar to the harmonic heat flow.The following is the arrangement of this thesis, which contains the contents of our research on this model, the main difficulties and our contributions to overcome these difficulties.In chapter1, we first recall the history of the research on the Ericksen-Lesile model in mathematics, then illustrate the main ideas of the proof of the well-posedness results as well as the contributions of this thesis. We also discuss the new features and associ-ated mathematical difficulties of the problems under consideration.Chapter2gives some basic materials and inequalities which will be used fre-quently in the thesis.Chapter3is concerned with the compressible liquid crystals model. Recently, Liu et al.(see [71,107]) have obtained the global weak solutions and the long behavior of the G-L model. However, it is hard and still an open question to get similar results for the non-G-L model. In this thesis, we prove the existence of the local strong solu-tions and the global strong solutions with small initial data. One difficulty is the initial density allows vacuum and the momentum equation becomes a degenerate parabolic-elliptic equation. To overcome it, the author approximate the initial density by a pos-itive initial density. Another difficulty is brought by the coupling term. Based on the observation on the scaling of u and d, we establish a two-variable iteration scheme and prove that the iterative sequence converges to the strong solutions of the equations. This technique is available not only for the proof of the local strong solutions, but also for the proof of global strong solutions with small initial data. In addition, using the same process, we can prove that the non-G-L model has a local strong solution.Chapter4establishes a blowup criterion for the compressible G-L model. The blowup criterion only relies on the speed u, but not the density p or the director d.Chapter5is devoted to a simple incompressible Ericksen-Lesile model in two dimensional bounded domain with Dirichlet boundary conditions. Our result is new for the non-G-L model. The previous work mainly concentrated in the G-L model, such as Lin-Liu’s papers (see [65,66]). The model considered in this chapter contains most of difficulties of the original model, such as the term|▽d|2d which has second order growth of gradient, the nonlinear pressure term div ((▽d+|▽d|2d)(?)d) and the transport terms d·▽u,(dT Ad) d. Moreover, compared with the periodic boundary conditions, the Dirichlet boundary conditions brings extra technical difficulty because it is inability of distribution. To overcome these above difficulties, the contributions of our thesis are:obtaining a local solution of an ODE system in Sobolev space and using a modified Galerkin method to get approximate solutions, adding a new term to the higher energy law to establish a prior estimates, finding a new elliptic operator to get u and d higher regularity in space by Stokes estimates. Eventually, the author obtains the strong solutions of the model in (0,∞). In addition, one can obtain the same results for the Neumann boundary conditions of d.As an appendix, chapter6summarizes the work of physicists on the continuum theory for liquid crystals and establishes the Ericksen-Lesile model to describe the motion of liquid crystals.