### The Ginzburg-Landau functional with vanishing magnetic field (after K. Attar and Helffer-Kachmar)

*Bernard Helffer*

*ProfesseurDépartement de Mathématiques, Université Paris-Sud 11Laboratoire Jean Leray, Universit e de Nantes*

We study the infimum of the Ginzburg-Landau functional in the case with a vanishing external magnetic field in a two dimensional simply connected domain. We obtain an energy asymptotics which is valid when the Ginzburg-Landau parameter is large and the strength of the external field is comparable with the third critical field. Compared with the known results when the external magnetic field does not vanish, we show in this regime a concentration of the energy near the zero set of the external magnetic field.

### 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.

### Counter-examples to Strong Diamagnetism

*Søren Fournais*

*Professor of 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 *bβ*, 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)* if *b1>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.

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

*Apala Majumdar*

*ProfessorDepartment of Mathematical SciencesUniversity 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).

### On the Dynamical Q-tensor Models of Liquid Crystals

*Shijin DING*

*Dean & ProfessorDepartment of MathematicsSouth China Normal University*

In this talk, we first introduce the 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.

### Plateau Problems in Singular Spaces

*Robert Hardt*

*ProfessorDepartment of Mathematics Rice University *

Many extremal variational problems involve a geometric constraint where each competing object or some part of it is required to lie in some fixed set. For example , a liquid crystal may be viewed as a map of a spatial region into the unit 2-sphere. A Plateau problem may require that each surface competing for least area lies in a fixed set A (Thus the complement of A is an obstacle). Also the boundary of the surface may be only partially fixed with the remaining part free to range in some subset B. In general dimensions, these problems may, following [Federer-Fleming,1960], be studied with classes of currents.

The existence and regularity properties of the minimizing currents depend on analytic properties of the sets A and B. We will discuss, with some general theorems and several examples, the role of smoothness and isoperimetric properties of the sets. We will refer to joint works with T De Pauw, W. Pfeffer, and Q. Funk.

### Energy-Minimizing Nematic Elastomers

*Patricia Bauman *

*Professor of MathematicsDepartment of MathematicsPurdue University*

We prove weak lower semi-continuity and existence of energy-minimizers for a free energy describing stable deformations and the corresponding director configuration of an incompressible nematic liquid-crystal elastomer subject to physically realistic bounda ry conditions. The energy is a sum of the trace formula developed by Warner, Terentjev and Bladon (coupling the deformation gradient and the director field) and the bulk term for the director with coefficients depending on temperature. A key step in our analysis is to prove that the energy density has a convex extension to non-unit length director fields.

Our results apply to the setting of physical experiments in which a thin incompressible elastomer in R^3 is clamped on its sides and stretched perpendicular to its initial director field, resulting in shape-changes and director re-orientation.

### On Nematic Liquid Crystal Flows in Dimension Three

*Changyou WANG*

*Professor of MathematicsUniversity of Kentucky*

In this talk, I will discuss a simplified Ericksen-Leslie system modeling the hydrodynamic flow of nematic liquid crystals in dimension three, and present a new result on the existence of global weak solutions. This is a joint work with Professor Fanghua Lin.

### Global and Local Stability of The Normal State of Superconductors under the Effect of Strong Electric Current

*Yaniv Almog*

*ProfessorDepartment of MathematicsLouisiana State University*

Consider a superconducting wire whose temperature is lower than the critical one. When the current through the wire exceeds some critical value, it is well known from experimental observation that the wire becomes resistive, behaving like a normal metal. We prove that the time-dependent Ginzburg-Landau model anticipates this behavior, and obtain upper bound for the critical current. The bounds are obtained in terms of the resolvent of the linearized elliptic operator in ${\mathbb R}^2$ and ${\mathbb R}^2_+$. We then relate this problem to some spectral analysis of a more general class of non-selfadjoint operators.

### Global Well-posedness of Incompressible Elastodynamics in 2D

*Zhen LEI*

*Professor of MathematicsSchool of Mathematical SciencesFudan University*

In this talk I will report our recent result on the global wellposedness of classical solutions to system of incompressible elastodynamics in 2D. The system is revealed to be inherently strong linearly degenerate and automatically satisfies a strong null condition, due to the isotropic nature and the incompressible constraint.

### Logarithmic Interaction Energy for Infinitely Many Points in the Plane, Coulomb Gases and Weighted Fekete Sets

*Etienne Sandier*

*ProfessorDépartement de MathématiquesUniversité Paris 12 Val de Marne*

To a configuration of infinitely many points in the plane, one can associate an energy describing the coulombian interaction of positive charges placed at these points with a negatively charged uniform background. I will describe results obtained in collaboration with S.Serfaty which give some basic properties of this energy and links it to superconducting vortices, log-gases and weighted Fekete sets. I will also describe criteria obtained in collaboration with Y.Ge which insure that this energy is finite.

### Mathematical Analysis of Liquid Crystal Models in Biology

*Maria-Carme Calderer*

*Professor of MathematicsSchool of MathematicsUniversity of Minnesota*

In this lecture, I present and analyze models of anisotropic crosslinked polymers employing tools from the theories of nematic liquid crystals and liquid crystal elastomers. The anisotropy of these systems stems from the presence of rigid-rod molecular units in the network. Energy functionals of compressible, incompressible elastomers as well as rod-fluid networks are addressed. The theorems on the minimization of the energies combine methods of isotropic nonlinear elasticity with the theory of lyotropic liquid crystals. A main feature of the systems is the coupling between the Eulerian Landau-de Gennes liquid crystal energy and the Lagrangian anisotropic elastic energy of deformation. Predictions of cavities in the minimizing configurations follow as a result of the nature of the coupling. I will also present a mixed finite element analysis of the incompressible elastomer.

We apply the theory to the study of phase transitions in networks of rigid rods, in order to model the behavior of actin filament systems found in the cytoskeleton of the interstitial tissue and in the inner cell.

The phase transition behavior depends on geometric and physical parameters of the network as well as on the aspect ratio, length and density of the rigid groups. In particular, we focus on the formation of chevron structures in the case that the aspect ratio of the rods is small.

The results show good agreement with the molecular dynamics experiments reported in the literature as well as with laboratory experiments.

Finally, we address the problem of polymer encapsulation to study configurations of bacteriophages.

### Regularity of Minimizers to a Constrained $Q$-tensor Energy for Liquid Crystals

*Daniel Phillips*

*Professor of MathematicsDepartment of MathematicsPurdue University*

We investigate minimizers defined on a two-dimensional domain for the Maier--Saupe energy used to characterize nematic liquid crystal configurations. The energy density for the model is singular so as to constrain the competing states to take on physical values. We prove that minimizers are regular and in certain cases we are able to use this regularity to prove that minimizers must take on values strictly within the physical regime. This work is joint with Patricia Bauman.

### Variational Problems in Smectic Liquid Crystals

*Jinhae Park*

*ProfessorDepartment of MathematicsChungnam National University*

In this talk, we will discuss a brief introduction to the governing energy functional for smectic liquid crystals including Chen-Lubensky energy terms. We then talk about existence of minimizers for Boundary Value Problems. Most part of this talk is from a joint work with P. Bauman and D. Phillips.

### Well-posedness and Stability of a Hydrodynamic System Modeling Vesicle and Fluid Interactions

*Hao WU*

*Associate Professor in Applied MathematicsSchool of Mathematical SciencesFudan University*

In this talk, we will discuss a hydrodynamic system modeling the deformation of vesicle membranes in incompressible viscous fluids. The system consists of the Navier-Stokes equations coupled with a fourth order phase-field equation. In the three dimensional case, we prove the existence/uniqueness of local strong solutions for arbitrary initial data as well as global strong solutions under the large viscosity assumption. We also establish some regularity criteria in terms of the velocity for local smooth solutions. Finally, we investigate the stability of the system near local minimizers of the elastic bending energy.

### On Ground States of Spin-1 Bose-Einstein Condensates w/o external magnetic field

*I-Liang Chern*

*Professor Department of Mathematics National Taiwan University*

In this talk, I will first give a brief introduction to the spinor Bose-Einstein condensates (BECs). Then I will present two recent results, one is numerical, the other is analytical for spinor BECs w/o uniform external magnetic field. In the numerical study of spinor BECs, a pseudo-arclength continuation method (PACM) was proposed for investigating the ground state patterns and phase diagrams of the spin-1 Bose-Einstein condensates under the influence of a homogeneous magnetic field. Two types of phase transitions are found. The first type is a transition from a two-component (2C) state to a three-component (3C) state. The second type is a symmetry breaking in 3C state. After that, a phase separation of the spin component occurs. In the semi-classical regime, these two phase transition curves are gradually merged.

In the analytical study, the ground states of spin-1 BEC are characterized. First, we present the case when there is no external magnetic field. For ferromagnetic systems, we show the validity of the so-called single-mode approximation (SMA). For antiferromagnetic systems, there are two subcases. When the total magnetization M≠0, the corresponding ground states have vanishing zeroth (*mF* = 0) components (so call 2C state), thus are reduced to two-component systems. When M=0, the ground states are also reduced to the SMA, and there are one-parameter families of such ground states. Next, we study the case when an external magnetic field is applied. It is shown analytically that, for antiferromagnetic systems, there is a phase transition from 2C state to 3C state as the external magnetic field increases. The key idea in the proof is a redistribution of masses among different components, which reduces kinetic energy in all situations, and makes our proofs simple and unified. The numerical part is a joint work with Jen-Hao Chen and Weichung Wang, whereas the analytical part is jointly with Liren Lin.

### New PNP Type Systems for Ionic Liquids

*Tai-Chia LIN*

*ProfessorDepartment of MathematicsNational Taiwan University*

To describe ionic liquids with finite size effects involving different ion radii and valences, we derive new PNP (Poisson-Nernst-Planck) type systems and develop mathematical theorems for these systems. Symmetry and non-symmetry breaking conditions are represented by their coupling coefficients. When non-symmetry breaking condition holds true, we prove the existence theorems of solutions of new PNP type systems. On the other hand, when symmetry breaking condition holds true, two steady state solutions can be found and the excess currents (due to steric effects) associated with these two steady state solutions are derived and expressed as two distinct formulas. Our results indicate that new PNP type systems may become a useful model to study ionic liquids and related topics of liquid crystals.

### Energetic Variational Approaches: General Diffusion and Stochastic Process

*Chun LIU*

*Professor of MathematicsDepartment of MathematicsPennsylvania State University*

In this talk, I will explore the variational structures in some specific types of non-ideal diffusion and stochastic processes. In particular, we will focus on those with nonlocal interactions and couplings.