Jun
15
4:30pm 4:30pm

On the Dynamical Q-tensor Models of Liquid Crystals

Shijin DING

Dean & Professor
Department of Mathematics
South China Normal University

In this talk, we first introduce the models of nematic liquid crystals and the known results about the models. Then, we focus on the dynamical Q-tensor model, that is, Beris-Edward model. For this model, we prove the global existence of weak solutions, the global existence of strong solutions with large viscosity, and the weak-strong uniqueness. In our discussions, the Landau-de Gennes functional takes a general form in which we only assume L_5=0, L_1>0 and L_2+L_3>0.

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Jun
15
3:30pm 3:30pm

Stability of Radial Solutions in the Landau-de Gennes Theory: Interplay between Temperature and Geometry

Apala Majumdar

Professor
Department of Mathematical Sciences
University of Bath

Nematics liquid crystals are anisotropic liquids with long-range orientational ordering, making them popular working materials for optical applications. The study of static nematic equilibria poses challenging questions in the calculus of variations and theory of partial differential equations. We study two stability problems for the prototypical radial-hedgehog solution within the Landau-de Gennes theory for nematics. The radial-hedgehog solution is an example of a uniaxial nematic equilibrium with an isotropic defect core, analogous to a degree +1-vortex solution in the Ginzburg-Landau theory of superconductivity. The first problem concerns the radial-hedgehog solution in a spherical droplet with radial boundary conditions, for low temperatures below the nematic-isotropic transition temperature.

We prove that an arbitrary sequence of Landau-de Gennes minimizers converges strongly (in W^{1,2}) to the radial-hedgehog solution in the low-temperature limit. We use the celebrated division trick for superconductivity, blow-up techniques for the singularity profile and energy estimates to show that the radial-hedgehog solution is the unique physically relevant uniaxial equilibrium in the low-temperature limit. We then compute the second variation of the Landau-de Gennes energy about the radial-hedgehog solution and demonstrate its instability with respect to higher-dimensional biaxial perturbations, for sufficiently low temperatures. We conclude that Landau-de Gennes minimizers on a spherical droplet, with radial anchoring, are always biaxial for sufficiently low temperatures.

The second problem concerns a punctured spherical droplet with radial boundary conditions. We show that the radial-hedgehog solution is locally stable for all temperatures below the nematic-isotropic transition temperature on a punctured droplet. We adapt methods from [1], [2] and use convexity-based properties of the Landau-de Gennes energy to prove that the radial-hedgehog solution is, in fact, the unique global energy minimizer in two different asymptotic limits: the vanishing elastic constant limit and the low-temperature limit, in contrast to the instability result for a spherical droplet above.

This is joint work with Duvan Henao, Adriano Pisante and Mythily Ramaswamy.

[1]  A. Majumdar, A. Zarnescu, Landau-de Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond, Arch. Rat. Mech. Anal., 196 (2010), no. 1, 227-280.
[2]  F.H.Lin and C.Liu, Static and Dynamic Theories of Liquid Crystals, Journal of Partial Differential Equations, Vol 14, No. 4, 289-330 (2001).

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Jun
15
2:00pm 2:00pm

Counter-examples to Strong Diamagnetism

Søren Fournais

Professor
Department of Mathematical Sciences
University of Aarhus

Consider a Schrödinger operator with magnetic field B(x) in 2-dimensions. The classical diamagnetic inequality implies that the ground state energy, denoted by λ1(B) , with magnetic field is higher than the one without magnetic field. However, comparison of the ground state energies for different non-zero magnetic fields is known to be a difficult question. We consider the special case where the magnetic field has the form , where b is a (large) parameter and β(x) is a fixed function. One might hope that monotonicity for large field holds, i.e. that λ1(b1β)>λ1(b2β) ifb1>b2 are sufficiently large. We will display counterexamples to this hope and discuss applications to the theory of superconductivity in the Ginzburg-Landau model. This is joint work with Mikael Persson Sundqvist.

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Jun
15
1:00pm 1:00pm

Defects of Liquid Crystals

Pingwen ZHANG

Vice Dean and Changjiang Professor
Department of Scientific & Engineering Computing (DSEC) 
School of Mathematical Sciences (SMS) 
Center for Computational Science & Engineering (CCSE) 
Peking University (PKU)

Defects in liquid crystals (LCs) are of great practical and theoretical importance. Recently there is a growing interest in LCs materials under topological constrain and/or external force, but the defects pattern and dynamics are still poorly understood. We investigate three-dimensional spherical droplet within the Landau-de Gennes model under different boundary conditions. When the Q-tensor is uniaxial, the model degenerates to vector model (Oseen-Frank), but Q-tensor model is superior to vector model as the former allows biaxial in the order parameter. Using numerical simulation, a rich variety of defects pattern are found, and the results suggest that, line disclinations always involve biaxial, or equivalently, uniaxial only admits point defects. Then we believe that Q-tensor model is essential to include the disclinations line which is a common phenomena in LCs. The mathematical implication of this observation will be discussed in this talk.

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Jun
15
10:00am10:00am

One Dimensional Rectifiable Varifolds and Some Applications

Robert Hardt

Professor
Department of Mathematics
Rice University

Varifolds were originally introduced to describe various 2 dimensional minimal surfaces and soap film models. A varifold is stationary in a region  U  if its first variation of its mass is zero under deformations supported in U. A stationary one-dimensional varifold may model a  spider-web (possibly of variable thickness) where the region  U  is the complement of the attaching points for the web. F.Almgren and W.Allard studied their regularity in 1976. After reviewing some previous applications of one dimensional varifolds, we discuss a new one involving Michell trusses. These are cost-optimal balanced structures consisting of bars and cables. Introduced in a 1904 paper of an economist A.G. Michell, they have been treated in the Mechanical Engineering  literature and in interesting papers by R.Kohn and G. Strang (1983) and by G.Bouchitte, W.Gangbo, and  P.Sepulcher (2008). There are many basic open questions about the location and structure of Michel trusses.

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