Nonlinear Analysis of Continuum Theories: Statics and Dynamics
Monday 8th April 2013 to Friday 12th April 2013
09:15 to 09:50  Registration  
09:50 to 10:00  Introduction  
10:00 to 11:00 
D Kinderlehrer (Carnegie Mellon University) Remarks about the Janossy effect
Light can change the orientation of a liquid crystal. This is the optical Freedericksz transition, discovered by Saupe. In the Janossy effect, the threshold intensity of the Freedericksz transition is dramatically resuced by the addition of a small amount of dye to the sample. We investigate the theory for this effect derived by E, Kosa, and PalffyMuhoray. Several themes come together, including molecular motors and MongeKantorovich mass transport. This is joint work with Michal Kowalczyk.


11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Analysis of DisclinationLine Defects in Liquid Crytals
We describe mathematical results and techniques of analysis on the structure of defects in thin nematic liquid crystals described by minimizers of the Landaude Gennes energy involving a tensorvalued order parameter with Dirichlet boundary conditions of nonzero degree. We prove that as the coefficient of the elasticity term tends to zero, a limiting uniaxial texture forms with a finite number of defects, all of degree 1/2 or 1/2. We also describe the location of defects and the limiting energy.


12:30 to 14:00  Lunch  
14:00 to 15:00 
Modeling the Dynamics of Liquid Crystalline Systems
A key characteristic of liquid crystals, exploited by many device applications, is their symmetry mandated responsivity. Describing the dynamics of the response of liquid crystalline systems to excitations is therefore of considerable interest and importance. Since orientational order is usually characterized in terms of an order parameter, the dynamic response of liquid crystals is often described in terms of the time evolution of the order parameter. A more general description, which gives more information yet is often simpler that the traditional approach, is the time evolution of the generalized density function. We describe a general procedure to obtain such a description, discuss its implementation and give several illustrative examples.


15:00 to 15:30  Afternoon Tea  
15:30 to 16:00 
Eigenvalue Constraints and Regularity of Qtensor NavierStokes Dynamics
If the Qtensor order parameter is interpreted as a normalised matrix of second moments of a probability measure on the unit sphere, its eigenvalues are bounded below by 1/3 and above by 2/3. This constraint raises questions regarding the physical predictions of theories which employ the Qtensor; it also raises analytical issues in both static and dynamic Qtensor theories of nematic liquid crystals.
John Ball and Apala Majumdar recently constructed a singular map on traceless, symmetric matrices that penalises unphysical Qtensors by giving them an infinite energy cost. In this talk, I shall discuss some mathematical results for a modified BerisEdwards model of nematic dynamics into which this map is built, including the existence, regularity and socalled `strict physicality' of its weak solutions.


16:00 to 16:30 
Field instabilities of Smectic A liquid crystals in 2D and 3D
We study the de Gennes free energy to describe the undulations instability in smectic A liquid crystals subjected to magnetic fields. If a magnetic field is applied in the direction parallel to the smectic layers, an instability occurs above a threshold magnetic field. When the magnetic field reaches this critical threshold, periodic layer undulations are observed. We prove the existence and stability of the solution to the nonlinear system of de Gennes model using $\Gamma$convergence method and bifurcation theory. Numerical simulations will be given near and well above the threshold. An efficient numerical scheme for some free energy containing the second order gradient will be presented. Undulation instabilities on three dimensional systems will be also discussed.
Coauthor: Carlos J. GarciaCervera (UCSB) 

16:30 to 17:00  Discussion chaired by FH Lin  
17:00 to 18:00  Drinks Reception 
09:00 to 10:00 
P Zhang (Peking University) The Small Deborah Number Limit of the DoiOnsager Equation to the EricksenLeslie Equation
We present a rigorous derivation of the EricksenLeslie equation starting from the DoiOnsager equation. As in the fluid dynamic limit of the Boltzmann equation, we first make the Hilbert expansion for the solution of the DoiOnsager equation. The existence of the Hilbert expansion is connected to an open question whether the energy of the EricksenLeslie equation is dissipated. We show that the energy is dissipated for the EricksenLeslie equation derived from the DoiOnsager equation. The most difficult step is to prove a uniform bound for the remainder in the Hilbert expansion. This question is connected to the spectral stability of the linearized DoiOnsager operator around a critical point. By introducing two important auxiliary operators, the detailed spectral information is obtained for the linearized operator around all critical points. However, these are not enough to justify the small Deborah number limit for the inhomogeneous DoiOnsager equation, since the elastic stress in the velocity equation is also strongly singular. For this, we need to establish a precise lower bound for a bilinear form associated with the linearized operator. In the bilinear form, the interactions between the part inside the kernel and the part outside the kernel of the linearized operator are very complicated. We find a coordinate transform and introduce a five dimensional space called the MaierSaupe space such that the interactions between two parts can been seen explicitly by a delicate argument of completing the square. However, the lower bound is very weak for the part inside the MaierSaupe space. In order to apply them to the error estimates, we have to analyze the structure of the singular terms and introduce a suitable energy functional. Furthermore, we prove the local wellposedness of the EricksenLeslie system, and the global wellposednss for small initial data under the physical constrain condition on the Leslie coefficients, which ensures that the energy of the system is dissipated. Instead of the GinzburgLandau approximation, we construct an approximate system with the dissipated energy based on a new formulation of the system.


10:00 to 11:00 
Tangent unitvector fields: nonabelian homotopy invariants, the Dirichlet energy and their applications in liquid crystal devices
We compute the infimum Dirichlet energy, E(H), of unitvector fields defined on an octant of the unit sphere, subject to tangent boundary conditions on the octant edges and of arbitrary homotopy type denoted by H. The expression for E(H) involves a new topological invariant – the spelling length – associated with the (nonabelian) fundamental group of the ntimes punctured twosphere. These results are then used for the modelling of the Post Aligned Bistable Nematic (PABN) device, designed by Hewlett Packard Laboratories. We provide analytic approximations for the experimentally observed stable equilibria in the PABN device and propose novel topological and geometrical mechanisms for bistability or multistability in prototype liquid crystal device geometries. This is joint work with Jonathan Robbins, Maxim Zyskin and Chris Newton.


11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
On stability of radial hedgehog in Landau  de Gennes model
We investigate stability of radially symmetric solutions in the context of Landau  de Gennes theory. It is well known that radial hedgehog is an unstable solution for low enough temperatures. We show that radial hedgehog is locally stable solution for temperatures close to isotropicnematic phase transition temperature.


12:30 to 14:00  Lunch  
14:00 to 15:00 
AD Zarnescu (University of Sussex) Eigenframe discontinuities, commutators and nematic defects
In the framework of De Gennes' Qtensor theory of nematics one can interpret defects as eigenframe discontinuities. A significant analytical difficulty related to understanding these discontinuities is due to the rather complicated relation between the multiparameter dependent matrices and their parametrized eigenvectors. We present necessary and sufficient criteria for determining eigenframe discontinuities (criteria expressed in terms of suitable commutators) as well as some consequences. This is joint work with Jonathan Robbins and Valeriy Slastikov.


15:00 to 15:30  Afternoon Tea  
15:30 to 16:00 
Wellposedness of a Coupled NavierStokes/Qtensor System
In this work, we show the existence and uniqueness of local strong solution for a coupled NavierStokes/Qtensor system on a bounded domain $\Omega\subset\mathbb{R}^3$ with Dirichlet boundary condition. One of the novelties brought in with respect to the existing literature consists in the fact that we deal with NavierStokes equation with variable viscosity. Concerning the methodology, we use an approximation method to handle the linearized system and the existence of solution to the nonlinear system is proved via a Banach's fixed point argument, based on the estimates on the lower order terms.


16:00 to 16:30 
Radial symmetry and biaxiality in nematic liquid crystals
We study the model problem of a nematic liquid crystal confined to a spherical droplet subject to radial anchoring conditions, in the context of the Landaude Gennes continuum theory. Based on the recent radial symmetry result by Millot & Pisante (J. Eur. Math. Soc. 2010) and Pisante (J. Funct. Anal. 2011) for the vectorvalued GinzburgLandau equations in threedimensional superconductivity theory, we prove that global Landaude Gennes minimizers in the class of uniaxial Qtensors converge, in the lowtemperature limit, to the radialhegdehog solution of the tensorvalued GinzburgLandau equations. Combining this with the result by Majumdar (Eur. J. App. Math. 2012) and by Gartland & Mkaddem (Phys. Rev. E. 1999) that the radialhedgehog equilibrium is unstable under biaxial perturbations, we obtain the nonpurely uniaxial character of global minimizers for sufficiently low temperatures.


16:30 to 17:00  Discussion chaired by OD Lavrentovich 
09:00 to 10:00 
An inverse problem arising from polarimetric measurements of nematic liquid crystals
This work is motivated by a polarimetric experiment (performed in HP Labs) were a thin slab of nematic liquid crystal was placed on a cylindrical mount, and illuminated by a focused polarized laser beam. As the slab is rotated and the polarimetric measurement data (the socalled Stokes parameters) varies with the angle of incidence. The object of this work was to determine what information on the dielectric permittivity of the liquid crystal could be retrieved from this data.


10:00 to 11:00 
Spontaneous flows and defect proliferation in active nematic liquid crystals
Active liquid crystals are nonequilibrium fluids composed of internally driven elongated units. Examples include mixtures of cytoskeletal filaments and associated motor proteins, bacterial suspensions, the cell cytoskeleton and even nonliving analogues, such as monolayers of vibrated granular rods. Due to the internal drive, these systems exhibit a host of nonequilibrium phenomena, including spontaneous laminar flow, large density fluctuations, unusual rheological properties, excitability, and low Reynolds number turbulence. In this talk I will review some of this phenomena and discuss new results on the dynamics and proliferation of topological defects in active liquid crystals. A simple analytical model for the defect dynamics will be shown to reproduce the key features of recent experiments in microtubulekinesin assemblies.


11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
P Biscari (Politecnico di Milano) Anisotropic elasticity and relaxation in nematic liquid crystals
As early as 1972, Mullen and coworkers showed experimentally that the director alignment of a nematic liquid crystal induces an anisotropic, frequency dependent sound speed in nematic liquid crystals. Similarly, Selinger and coworkers have studied a liquid crystal cell where the nematic molecules can be realigned by an ultrasonic wave, leading to a change in the optical transmission through the cell. The existing theoretical models for this acoustooptic effect propose a free energy that depends on the density gradient thus describing the nematic liquid crystal as a compressible second grade fluid. In this talk we will show that that the angular dependence of the sound speed can be easily reproduced by introducing a simple anisotropic term in the stress tensor, thus providing a simpler firstgrade model for the acoustooptic effect. The simplest term is nonhyperelastic, but we show that it can be interpreted as the quasiincompressible approximation of an elastic term which couples the director orientation with the strain. More interestingly, the frequency dependence of the anisotropic sound speed can be recovered by assuming an irreversible relaxation of the reference configuration with respect to which the strain is measured.


12:30 to 13:00  Discussion chaired by E Virga  
13:00 to 14:30  Lunch  
14:30 to 17:00  Free 
09:00 to 10:00 
Derivation of the Balance Laws for Liquid Crystals using Statistical Mechanics
I will outline how one can derive the continuumlevel balances for liquid crystals using statistical mechanics. I will start by considering a discrete system of rigid rods, which motivates an appropriate state space. A probability function is then introduced that satisfies the Liouville equation. This equation serves as the starting point for the derivation of all of the continuumlevel balances. The terms appearing in the derived balances, some being nonstandard, are interpreted as expected values.


10:00 to 11:00 
N Walkington (Carnegie Mellon University) Numerical Approximation of the Ericksen Leslie Equations
The Ericksen Leslie equations model the motion of nematic liquid crytaline fluids. The equations comprise the linear and angular momentum equations with nonconvex constraints on the kinematic variables. These equations possess a Hamiltonian structure which reveals the subtle coupling of the two equations, and a delicate balance between inertia, transport, and dissipation. While a complete theory for the full nonlinear system is not yet available, many interesting subcases have been analyzed.
This talk will focus on the development and analysis of numerical schemes which inherit the Hamiltonian structure, and hence stability, of the continuous problem. In certain situations compactness properties of the discrete solutions can be established which guarantee convergence of schemes. 

11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Liquid crystal phases of biological networks: models and analysis
Cytoskeletal networks consist of rigid, rodlike actin protein units jointed by flexible crosslinks, presenting coupled orientational and deformation effects analogous to liquid crystal elastomers. The alignment properties of the rigid rods influence the mechanical response of the network to applied stress and deformation, affecting functionality of the systems. Parameters that characterize these networks include the aspect ratio of the rods and the average length of the crosslinks, with a large span of parameter values found across invivo networks. For instance, cytoskeletal networks of red blood cells have very large linkers and small rod aspect ratio, whereas those of cells of the outer hair of the ear have large aspect ratio and short linkers favoring well aligned nematic, in order to achieve optimal sound propagation. We propose a class of free energy densities consisting of the sum of polyconvex functions of the anisotropic deformation tensor and the Landaude Gennes energy of lyotropic liquid crystals. The growth conditions of the latter, with respect to the rod density and the nematic order tensor at the limit of the minimum eigenvalue 1/3 are essential to recover the limiting deformation map from the minimizing sequences of the anisotropic deformation gradient. We consider a bulk free energy density encoding properties of the rod and the network based on the LopatinaSelinger construction for the MaierSaupe theory. We then analyze the phase transition behavior under uniform expansion, biaxial extension and shear deformation, showing that the nematicisotropic transition may be accompanied by a change of volume, which manifests itself in the nonconvexity of the stressstrain relation. We also account for the fact that invivo networks are found in the gel state. We conclude with some remarks on the roles of active elements in the model.


12:30 to 14:00  Lunch  
14:00 to 15:00 
Some results on the existence of solutions to the EricksenLeslie system
The EricksenLeslie theory describes the dynamic flow of liquid crystals. In this talk, we will discuss global existence of solutions of the EricksenLeslie system for a general OseenFrank model in 2D. We also discuss some new results on the local existence, uniqueness and the blow up criterions of strong solutions to the EricksenLeslie system in 3D for the general OseenFrank model in 3D.


15:00 to 15:30  Afternoon Tea  
15:30 to 16:30 
C Wang (University of Kentucky) Some recent results on analysis of nematic liquid crystal flows
In this talk, I will survey some recent works on the analysis of a simplified version of EricksenLeslie equation modeling the hydrodynamic motions of nematic liquid crystals.


16:30 to 17:00  Discussion chaired by C Zannoni  
19:00 to 22:00  Drinks and Conference Dinner at Oriel College 
09:00 to 10:00 
Mechanical response and microstructures in liquid crystal elastomers: small vs large strain theories
In this talk, we will present recent theoretical and numerical results for models of nematic elastomers within the small strain approach.
While strains exhibited by nematic elastomers are usually large, there are cases where this is not so, and the early modeling approaches were using this framework. In fact, the main reason for the developing small strain theories for nematic elastomers is the clear geometric structure of the resulting energy landscape. We will exploit this structure to discuss material instabilities and stressstrain diagrams, and to suggest possible generalizations to more realistic models. 

10:00 to 11:00 
Polar Active Liquid Crystals : microscopics, hydrodynamics and rheology
Colonies of swimming bacteria, mixtures of cytoskeletal protein filaments and motor proteins, and vibrated granular rods are examples of active systems composed of interacting units that consume energy and collectively generate motion and mechanical stresses. Due to their elongated shape, active particles can exhibit orientational order at high concentration and have been likened to ``living liquid crystals". Their rich collective behavior includes nonequilibrium phase transitions and pattern formation on mesoscopic scales. I will describe and summarise recent theoretical results characterising the behaviour of such soft active systems.


11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
D Phillips (Purdue University) Analysis of defects in minimizers for a planar Frank energy
Smectic C* liquid crystal films are modeled with a relaxed Frank energy, \begin{equation*} \int_\Omega\Big( k_s(\text{div}\, u)^2 + k_b(\text{curl}\, u)^2 + \frac{1}{2\epsilon^2}(1  u^2)^2 \Big)\, dx . \end{equation*} Here $k_s$ and $k_b$ represent the two dimensional splay and bend moduli for the film respectively with $k_s, k_b > 0$, $\Omega$ is a planar domain, and $u$ is an $\mathbb{R}^2$valued vector field with fixed boundary data having degree $d>0$. We study the limiting pattern for a sequence of minimizers $\{u_\epsilon\}$ as $\epsilon\to 0$. We prove that the pattern contains $d$, degree one defects and that it has a either a radial or circular asymptotic form near each defect depending on the relative values of $k_s$ and $k_b$. We further characterize a renormalized energy for the problem and show that it is minimized by the limit. This is joint work with Sean ColbertKelly.


12:30 to 12:45  Discussion chaired by J Ball 