Abstract:
In this paper we investigate the three dimensional general Ericksen-Leslie (E--L) system with Ginzburg-Landau type approximation modeling nematic liquid crystal flows. First, by overcoming the difficulties from lack of maximum principle for the director equation and high order nonlinearities for the stress tensor, we prove existence of global-in-time weak solutions under physically meaningful boundary conditions and suitable assumptions on the Leslie coefficients, which ensures that the total energy of the E--L system is dissipated. Moreover, for the E--L system with periodic boundary conditions, we prove the local well-posedness of classical solutions under the so-called Parodi's relation and establish a blow-up criterion in terms of the temporal integral of both the maximum norm of the curl of the velocity field and the maximum norm of the gradient of the liquid crystal director field.

Abstract:
In this paper, we first establish the regularity theorem for suitable weak solutions to the Ericksen-Leslie system in dimensions two. Building on such a regularity, we then establish the existence of a global weak solution to the Ericksen-Leslie system in $\mathbb R^2$ for any initial data in the energy space, under the physical constraint conditions on the Leslie coefficients ensuring the dissipation of energy of the system, which is smooth away from at most finitely many times. This extends earlier works by Lin,Lin, and Wang on a simplified nematic liquid crystal flow in dimensions two.

Abstract:
Liquid crystal display panels subjected to tactile force will show ripple propagation on screens. Tactile forces change tilt angles of liquid crystal molecules and alter optical transmission so as to generate ripple on screens. Based on the Ericksen-Leslie theory, this study investigates ripple propagation by dealing with tilt angles of liquid crystal molecules. Tactile force effects are taken into account to derive the molecule equation of motion for liquid crystals. Analytical results show that viscosity, tactile force, the thickness of cell gap, and Leslie viscosity coefficient lead to tilt angle variation. Tilt angle variations of PAA liquid crystal molecules are sensitive to tactile force magnitudes, while those of 5CB and MBBA with larger viscosity are not. Analytical derivation is validated by numerical results. 1. Introduction Many physical phenomena exhibited by the nematic phase liquid crystals (LC), such as unusual flow properties or the LC response to electric and magnetic fields, can be studied by treating LC as a continuous medium. Ericksen and Leslie [1–3] formulated general conservation laws and constitutive equations describing the dynamic behavior. Other continuum theories have been proposed, but it turns out that the Ericksen-Leslie theory is the one that is most widely used in discussing the nematic state. Based on the continuum theory of Ericksen-Leslie, this study constructs a theoretical model in order to investigate the tilt angle variation of LC subjected to both electric field and tactile force. The continuum theory for the nematic state flow is established in two dynamical equations—conservation laws and constitutive equations. Both equations are coupled with each other due to the properties of flow and the director of liquid crystal molecules. Brochard et al. [4] investigated transient distortions in a nematic film by increasing or decreasing of the magnetic field. Lei et al. [5] used the Ericksen-Leslie equation to describe soliton propagation in nematic liquid crystals under shear. Leslie [6] presented a concise but clear derivation of continuum equations commonly employed to describe static and dynamic phenomena in nematic liquid crystals. Gleeson et al. [7] proposed a two-dimensional theory in which the excitations are fronts between distinct solutions of the steady-state Ericksen-Leslie equations. Lin and Liu [8] use the Ericksen-Leslie equation to describe the flow of liquid crystal material and prove the global existence of weak solutions. De Andrade Lima and Rey [9] proposed computational modeling of the steady

Abstract:
In this paper, we consider Cauchy problem of simplified Ericksen-Leslie system in dimension three. We establish the unique existence of global smooth solution under some nonlinear conditions on initial data. However, we do not need small conditions on initial data.

Abstract:
We consider the weak solution of the simplified Ericksen-Leslie system modeling compressible nematic liquid crystal flows in $\mathbb R^3$. When the initial data is small in $L^2$ and initial density is positive and essentially bounded, we first prove the existence of a global weak solution in $\mathbb R^3$. The large-time behavior of a global weak solution is also established.

Abstract:
We consider weak solutions to a two-dimensional simplified Ericksen-Leslie system of compressible flow of nematic liquid crystals. An initial-boundary value problem is first studied in a bounded domain. By developing new techniques and estimates to overcome the difficulties induced by the supercritical nonlinearity in the equations of angular momentum on the direction field, and adapting the standard three-level approximation scheme and the weak convergence arguments for the compressible Navier-Stokes equations, we establish the global existence of weak solutions under a restriction imposed on the initial energy including the case of small initial energy. Then the Cauchy problem with large initial data is investigated, and we prove the global existence of large weak solutions by using the domain expansion technique and the rigidity theorem, provided that the second component of initial data of the direction field satisfies some geometric angle condition.

Abstract:
For any bounded smooth domain $\Omega\subset\mathbb R^3$, we establish the global existence of a weak solution $u:\Omega\times (0,+\infty)\to\mathbb R^3\times\mathbb S^2$ of the initial-boundary value (or the Cauchy) problem of the simplified Ericksen-Leslie system (1.1) modeling the hydrodynamic flow of nematic liquid crystals for any initial and boundary (or Cauchy) data $(u_0. d_0)\in {\bf H}\times H^1(\Omega,\mathbb S^2$), with $d_0(\Omega)\subset\mathbb S^2_+$ (the upper hemisphere). Furthermore, ($u,d$) satisfies the global energy inequality (1.4).

Abstract:
This paper concerns the uniqueness of weak solutions to the Cauchy problem to the Ericksen-Leslie system of liquid crystal models in $\mathbb R^2$, with both general Leslie stress tensors and general Oseen-Frank density. It is shown here that such a system admits a unique weak solution provided that the Frank coefficients are close to some positive constant, which solves an interesting open problem. One of the main ideas of our proof is to perform suitable energy estimates at level one order lower than the natural basic energy estimates for the Ericksen-Leslie system.

Abstract:
In this paper, we prove the existence and uniqueness of local strong solutions of the hydrodynamics of nematic liquid crystals system under the initial data satisfying a natural compatibility condition. Also the global strong solutions of the system with small initial data are obtained.

Abstract:
In the first part of this paper, we establish global existence of solutions of the liquid crystal (gradient) flow for the well-known Oseen-Frank model. The liquid crystal flow is a prototype of equations from the Ericksen-Leslie system in the hydrodynamic theory and generalizes the heat flow for harmonic maps into the 2-sphere. The Ericksen-Leslie system is a system of the Navier-Stokes equations coupled with the liquid crystal flow. In the second part of this paper, we also prove global existence of solutions of the Ericksen-Leslie system for a general Oseen-Frank model in $\Bbb R^2$.