Jun
18
4:30pm 4:30pm

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

Hao WU

Associate Professor in Applied Mathematics
School of Mathematical Sciences
Fudan 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.

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Jun
18
3:30pm 3:30pm

Variational Problems in Smectic Liquid Crystals

Jinhae Park

Professor
Department of Mathematics
Chungnam 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.

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Jun
18
2:00pm 2:00pm

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

Daniel Phillips

Professor of Mathematics
Department of Mathematics
Purdue 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.

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Jun
18
1:00pm 1:00pm

Mathematical Analysis of Liquid Crystal Models in Biology

Maria-Carme Calderer

Professor of Mathematics
School of Mathematics
University 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. 

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Jun
18
10:00am10:00am

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

Etienne Sandier

Professor
Département de Mathématiques
Université 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.

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