Zhen LEI

Professor of Mathematics
School of Mathematical Sciences
Fudan 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.

Yaniv Almog

Professor
Department of Mathematics
Louisiana 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.

Changyou WANG

Professor of Mathematics
University 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.

Patricia Bauman

Professor of Mathematics
Department of Mathematics
Purdue 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 boundary 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.

Bernard Helffer

Professeur
Département de Mathématiques, Université Paris-Sud 11
Laboratoire Jean Leray, Universite 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.