We study the bifurcation diagram of the Onsager free-energy func- tional for liquid crystals with orientation parameter on the sphere in three dimensions. In particular, we concentrate on a very general class of two- body interaction potentials including the Onsager kernel. The problem is reformulated as a non-linear eigenvalue problem for the kernel operator, and a general method to find the corresponding eigenvalues and eigenfunctions is presented. Our main tools for this analysis are spherical harmonics and a special algorithm for computing expansions of products of spherical harmon- ics in terms of spherical harmonics. We find an explicit expression for the set of all bifurcation points. Using a Lyapunov-Schmidt reduction, we derive a bifurcation equation depending on five state variables. The dimension of this state space is further reduced to two dimensions by using the rotational symmetry of the problem and the theory of invariant polynomials. On the basis of these result, we show that the first bifurcation occurring in case of the Onsager interaction potential is a transcritical bifurcation and that the corresponding solution is uniaxial. In addition, we prove some global proper- ties of the bifurcation diagram such as the fact that the trivial solution is the unique local minimiser for high temperatures, that it is not a local minimiser if the temperature is low, the boundedness of all equilibria of the functional and that the bifurcation branches are either unbounded or that they meet another bifurcation branch.
We study the relationship between Onsager's molecular theory and the Oseen-Frank theory for nematic liquid crystals. Under the molecular setting, we consider the free energy that includes the effects of nonlocal molecular interactions. By imposing the strong anchoring boundary condition on the second moment of the number density function, we prove the existence of global minimizers for the free energy. Moreover, when the re-scaled interaction distance tends to zero, the corresponding global minimizers will converge to a uniaxial distribution whose orientation is described by a minimizer of Ose…
We give new results of the phase transition of dilute colloidal solutions of rod-like molecules in dimension $D \geq 3$. For the low concentration of particles in a carrier fluid, we prove that the isotropic phase is the unique solution to the Doi-Onsager model with the general potential kernel. In addition, we present the regime of the bifurcation of nematic phases in the class of axially symmetric solutions. Our method is based on a generalization of the classical Leray-Schauder degree we developed for this problem.
On strictly starshaped domains of second kind we find natural sufficient conditions which allow the solution of two long standing open problems closely related to the mean field equation $\prl$ below. On one side we catch the global behaviour of the Entropy for the mean field Microcanonical Variational Principle ((MVP) for short) arising in the Onsager description of two-dimensional turbulence. This is the completion of well known results first established in Caglioti et al. Comm.Math.Phys. (1995). Among other things we find a full unbounded interval of strict convexity of the Entropy. On the …
Starting from Doi-Onsager equation for the liquid crystal, we first derive the Q-tensor equation by the Bingham closure. Then we derive the Ericksen-Leslie equation from the Q-tensor equation by taking the small Deborah number limit.
We study steady bifurcation for the coupled system body-liquid consisting of a sphere freely falling in a Navier-Stokes liquid under the action of gravity. In particular we show that, under the assumption that for the bifurcating solution the translational velocity of the sphere is parallel to the gravity, bifurcation takes place provided 1 is a simple eigenvalue of a suitable linear operator and the transversality property holds.
We provide sufficient conditions for the occurrence of time-periodic Hopf bifurcation for the coupled system constituted by a rigid sphere, $\mathscr S$, freely moving under gravity in a Navier-Stokes liquid. Since the region of flow is unbounded (namely, the whole space outside $\mathscr S$), the main difficulty consists in finding the appropriate functional setting where general theory may apply. In this regard, we are able to show that the problem can be formulated as a suitable system of coupled operator equations in Banach spaces, where the relevant operators are Fredholm of index 0. In s…
We investigate the effect of a strong magnetic field on a three dimensional smectic A liquid crystal. We identify a critical field above which the uniform layered state loses stability; this is associated to the onset of layer undulations. In a previous work, García-Cervera and Joo considered the two dimensional case and analyzed the transition to the undulated state via a simple bifurcation. In dimension n=3 the situation is more delicate because the first eigenvalue of the corresponding linearized problem is not simple. We overcome the difficulties inherent to this higher dimensional setting…
We study the solution landscape and bifurcation diagrams of nematic liquid crystals confined on a rectangle, using a reduced two-dimensional Landau--de Gennes framework in terms of two geometry-dependent variables: half short edge length $λ$ and aspect ratio $b$. First, we analytically prove that, for any $b$ with a small enough $λ$ or for a large enough $b$ with a fixed domain size, there is a unique stable solution that has two line defects on the opposite short edges. Second, we numerically construct solution landscapes by varying $λ$ and $b$, and report a novel X state, which emerges from …
We provide general sufficient conditions for branching out of a time-periodic family of solutions from steady-state solutions to the two-dimensional Navier-Stokes equations in the exterior of a cylinder. To this end, we first show that the problem can be formulated as a coupled elliptic-parabolic nonlinear system in appropriate function spaces. This is obtained by separating the time-independent averaged component of the velocity field from its "purely periodic" one. We then prove that time-periodic bifurcation occurs, provided the linearized time-independent operator of the parabolic problem …
In this paper, we consider a hydrodynamic $Q$-tensor system for nematic liquid crystal flow, which is derived from Doi-Onsager molecular theory by the Bingham closure. We first prove the existence and uniqueness of local strong solution. Furthermore, by taking Deborah number goes to zero and using the Hilbert expansion method, we present a rigorous derivation from the molecule-based $Q$-tensor theory to the Ericksen-Leslie theory.
From microscopic theory to macroscopic theory: a systematic study on static modeling for liquid crystals
LMAM & School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）
Using Landau-de Gennes theory to describe nematic order, we study a frustrated cell consisting of nematic liquid crystal confined between two parallel plates. We prove the uniqueness of equilibrium states for a small cell width. Letting the cell width grow, we study the behaviour of this unique solution. Restricting ourselves to a certain interval of temperature, we prove that this solution becomes unstable at a critical value of the cell width. Moreover, we show that this loss of stability comes with the appearance of two new solutions: there is a symmetric pitchfork bifurcation. This picture…
We study the effects of elastic anisotropy on the Landau-de Gennes critical points for nematic liquid crystals, in a square domain. The elastic anisotropy is captured by a parameter, $L_2$, and the critical points are described by three degrees of freedom. We analytically construct a symmetric critical point for all admissible values of $L_2$, which is necessarily globally stable for small domains i.e., when the square edge length, $λ$, is small enough. We perform asymptotic analyses and numerical studies to discover at least $5$ classes of these symmetric critical points - the $WORS$, $Ring^{…
Mathematisches Institut der Universität München（慕尼黑大学数学研究所）；Delft University of Technology（代尔夫特理工大学）
We discuss thermotropic nematic liquid crystals in the mean-field regime. In the first part of this article, we rigorously carry out the mean-field limit of a system of $N$ rod-like particles as $N\to\infty$, which yields an effective `one-body' free energy functional. In the second part, we focus on spatially homogeneous systems, for which we study the associated Euler-Lagrange equation, with a focus on phase transitions for general axisymmetric potentials. We prove that the system is isotropic at high temperature, while anisotropic distributions appear through a transcritical bifurcation as …
