# Workshop Programme

## for period 8 - 12 April 2013

### Nonlinear Analysis of Continuum Theories: Statics and Dynamics

8 - 12 April 2013

Timetable

Monday 8 April | ||||

09:15-09:50 | Registration | |||

09:50-10:00 | Introduction | |||

10:00-11:00 | Kinderlehrer, D (Carnegie Mellon University) |
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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 Palffy-Muhoray. Several themes come together, including molecular motors and Monge-Kantorovich mass transport. This is joint work with Michal Kowalczyk. |
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11:00-11:30 | Morning Coffee | |||

11:30-12:30 | Bauman, P (Purdue University) |
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Analysis of Disclination-Line 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 Landau-de Gennes energy involving a tensor-valued 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. |
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12:30-14:00 | Lunch | |||

14:00-15:00 | Palffy-Muhoray, P (Kent State University) |
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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. |
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15:00-15:30 | Afternoon Tea | |||

15:30-16:00 | Wilkinson, M (University of Oxford) |
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Eigenvalue Constraints and Regularity of Q-tensor Navier-Stokes Dynamics | ||||

If the Q-tensor 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 Q-tensor; it also raises analytical issues in both static and dynamic Q-tensor theories of nematic liquid crystals. John Ball and Apala Majumdar recently constructed a singular map on traceless, symmetric matrices that penalises unphysical Q-tensors by giving them an infinite energy cost. In this talk, I shall discuss some mathematical results for a modified Beris-Edwards model of nematic dynamics into which this map is built, including the existence, regularity and so-called `strict physicality' of its weak solutions. |
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16:00-16:30 | Joo, S (Old Dominion University) |
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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. Co-author: Carlos J. Garcia-Cervera (UCSB) |
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16:30-17:00 | Discussion chaired by F-H Lin | |||

17:00-18:00 | Drinks Reception |

Tuesday 9 April | ||||

09:00-10:00 | Zhang, P (Peking University) |
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The Small Deborah Number Limit of the Doi-Onsager Equation to the Ericksen-Leslie Equation | ||||

We present a rigorous derivation of the Ericksen-Leslie equation starting from the Doi-Onsager equation. As in the fluid dynamic limit of the Boltzmann equation, we first make the Hilbert expansion for the solution of the Doi-Onsager equation. The existence of the Hilbert expansion is connected to an open question whether the energy of the Ericksen-Leslie equation is dissipated. We show that the energy is dissipated for the Ericksen-Leslie equation derived from the Doi-Onsager 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 Doi-Onsager 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 Doi-Onsager 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 Maier-Saupe 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 Maier-Saupe 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 well-posedness of the Ericksen-Leslie system, and the global well-posednss 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 Ginzburg-Landau approximation, we construct an approximate system with the dissipated energy based on a new formulation of the system. |
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10:00-11:00 | Majumdar, A (University of Bath) |
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Tangent unit-vector fields: nonabelian homotopy invariants, the Dirichlet energy and their applications in liquid crystal devices | ||||

We compute the infimum Dirichlet energy, E(H), of unit-vector 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 n-times punctured two-sphere. 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. |
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11:00-11:30 | Morning Coffee | |||

11:30-12:30 | Slastikov, V (University of Bristol) |
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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 isotropic-nematic phase transition temperature. |
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12:30-14:00 | Lunch | |||

14:00-15:00 | Zarnescu, AD (University of Sussex) |
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Eigenframe discontinuities, commutators and nematic defects | ||||

In the framework of De Gennes' Q-tensor 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. |
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15:00-15:30 | Afternoon Tea | |||

15:30-16:00 | Liu, Y (Universität Regensburg) |
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Wellposedness of a Coupled Navier-Stokes/Q-tensor System | ||||

In this work, we show the existence and uniqueness of local strong solution for a coupled Navier-Stokes/Q-tensor 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 Navier-Stokes 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. |
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16:00-16:30 | Henao, DA (Pontificia Universidad Católica de Chile) |
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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 Landau-de 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 vector-valued Ginzburg-Landau equations in three-dimensional superconductivity theory, we prove that global Landau-de Gennes minimizers in the class of uniaxial Q-tensors converge, in the low-temperature limit, to the radial-hegdehog solution of the tensor-valued Ginzburg-Landau equations. Combining this with the result by Majumdar (Eur. J. App. Math. 2012) and by Gartland & Mkaddem (Phys. Rev. E. 1999) that the radial-hedgehog equilibrium is unstable under biaxial perturbations, we obtain the non-purely uniaxial character of global minimizers for sufficiently low temperatures. |
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16:30-17:00 | Discussion chaired by OD Lavrentovich |

Wednesday 10 April | ||||

09:00-10:00 | Capdeboscq, Y (University of Oxford) |
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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 so-called 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. |
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10:00-11:00 | Marchetti, MC (Syracuse University) |
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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 non-living 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 microtubule-kinesin assemblies. |
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11:00-11:30 | Morning Coffee | |||

11:30-12:30 | Biscari, P (Politecnico di Milano) |
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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 co-workers 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 acousto-optic 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 first-grade model for the acousto-optic effect. The simplest term is non-hyperelastic, but we show that it can be interpreted as the quasi-incompressible 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. |
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12:30-13:00 | Discussion chaired by E Virga | |||

13:00-14:30 | Lunch | |||

14:30-17:00 | Free |

Thursday 11 April | ||||

09:00-10:00 | Seguin, B (McGill University) |
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Derivation of the Balance Laws for Liquid Crystals using Statistical Mechanics | ||||

I will outline how one can derive the continuum-level 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 continuum-level balances. The terms appearing in the derived balances, some being nonstandard, are interpreted as expected values. |
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10:00-11:00 | Walkington, N (Carnegie Mellon University) |
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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 non-convex 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 sub-cases 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. |
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11:00-11:30 | Morning Coffee | |||

11:30-12:30 | Calderer, M-C (University of Minnesota) |
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Liquid crystal phases of biological networks: models and analysis | ||||

Cytoskeletal networks consist of rigid, rod-like 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 in-vivo 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 Landau-de 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 Lopatina-Selinger construction for the Maier-Saupe theory. We then analyze the phase transition behavior under uniform expansion, biaxial extension and shear deformation, showing that the nematic-isotropic transition may be accompanied by a change of volume, which manifests itself in the nonconvexity of the stress-strain relation. We also account for the fact that in-vivo networks are found in the gel state. We conclude with some remarks on the roles of active elements in the model. |
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12:30-14:00 | Lunch | |||

14:00-15:00 | Hong, M-C (University of Queensland) |
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Some results on the existence of solutions to the Ericksen-Leslie system | ||||

The Ericksen-Leslie theory describes the dynamic flow of liquid crystals. In this talk, we will discuss global existence of solutions of the Ericksen-Leslie system for a general Oseen-Frank model in 2D. We also discuss some new results on the local existence, uniqueness and the blow up criterions of strong solutions to the Ericksen-Leslie system in 3D for the general Oseen-Frank model in 3D. |
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15:00-15:30 | Afternoon Tea | |||

15:30-16:30 | Wang, C (University of Kentucky) |
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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 Ericksen-Leslie equation modeling the hydrodynamic motions of nematic liquid crystals. |
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16:30-17:00 | Discussion chaired by C Zannoni | |||

19:00-22:00 | Drinks and Conference Dinner at Oriel College |

Friday 12 April | ||||

09:00-10:00 | De Simone, A (SISSA) |
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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 stress-strain diagrams, and to suggest possible generalizations to more realistic models. |
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10:00-11:00 | Liverpool, T (University of Bristol) |
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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. |
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11:00-11:30 | Morning Coffee | |||

11:30-12:30 | Phillips, D (Purdue University) |
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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 Colbert-Kelly. |
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12:30-12:45 | Discussion chaired by J Ball |