## Spring 2014

Date Speaker Title/Abstract
27 Feb Valeriy Slastikov
University of Bristol
Point defects in liquid crystals
We study liquid crystal defects in cylindrical domains. We employ Landau-de Gennes theory and provide a simplified description of global minimizers of Landau-de Gennes energy under homeothropic boundary conditions at the lateral boundary of the cylinder. We also provide explicit solutions describing defects of various strength under Lyuksutov's constraint.
13 Mar Daniel Matthes
Technische Universität München
Convex Lyapunov functionals for non-convex gradient flows: two examples
Various diffusion equations can be written as a gradient flow, i.e., their solutions are curves of steepest descent (with respect to a suitable metric) in some energy landscape (of a suitable potential). If the potential is strictly convex in the considered metric, then one immediately obtains quantitative estimates on the speed of convergence of solutions towards equilibrium. In this talk, I will discuss two examples in which the potential is not convex, but still good estimates on the long-time asymptotics can be derived by variational methods. The first example (joint with S. Linisi and G. Savare) is a family of fourth order degenerate parabolic equations, which arise e.g. in models for lubrication. The second example (joint work with J. Zinsl) is a system of two nonlinear diffusion equations modeling the aggregation of bacteria.
27 Mar Nicolas Dirr
Cardiff University
Uniqueness for minimisers of a nonlocal random functional
We consider a small random perturbation of the energy functional acting on functions on $\mathbb{R}^n$, which consists of a double-well potential and the Dirichlet form associated with a fractional Laplacian. The unperturbed functional has two minimisers (under compact perturbations), the constant functions $+1$ and $-1$. For a range of parameters and dimensions we are able to show that the random perturbation, although of mean zero, enforces uniqueness of the minimiser. This is joint work with Enza Orlandi (Roma Tre).
10 Apr Euan Spence
University of Bath
Why analysts should care about numerical analysis of the Helmholtz equation
I have spent the last few years using results and techniques developed for the analysis of the Helmholtz equation to address certain questions about the numerical analysis of the Helmholtz equation. Many of these numerical analysis questions can be answered more-or-less immediately using the analysis results, however there are some questions (which I will discuss in this talk) for which the classical analysis results are not sufficient, and more work is required. (Note that absolutely no knowledge of any numerical analysis will be necessary to understand this talk.)
24 Apr Matthias Kurzke
University of Nottingham
The hydrodynamic limit of the parabolic Ginzburg-Landau equation
The Ginzburg-Landau functional serves as a model for the formation of vortices in various physical contexts. The natural gradient flow, the parabolic Ginzburg-Landau equation, converges in the limit of small vortex size and finite number of vortices to a point vortex system. Passing to the limit of many vortices in this ODE, one can derive a mean field PDE, similar to the passage from point vortex systems to the 2D Euler equations with vortex sheet initial data. In the talk, I will present quantitative estimates that allow us to directly connect the parabolic GL equation to the limiting mean field PDE. This is joint work with Daniel Spirn (University of Minnesota).
$\Gamma$-convergence analysis of discrete topological singularities: metastability and dynamics