Peter Embacher (Cardiff)
We consider a diffusive system that is microscopically driven by an underlying stochastic process. For some of these systems it has been shown that their macroscopical evolution can be described by a gradient flow along its entropy, as long as the corresponding thermodynamical metric is chosen conveniently. We propose a numerical method to extract this metric from given experimental data. The method was applied to zero-range-processes and proved successfull there.
Doaa Filali (Cardiff, UK)
Star-shaped sets play a signifcant role in analysis and PDEs. In particular convex sets are star-shaped with respect to the all interior points. In the Euclidean case there are several equivalent definitions for star-shaped sets. This is not true in more degenerate geometries. For example in the Heisenberg group we generalised several Euclidean definitions for starshapedness and they turned to be not equivalent. For example we looked at the definition of strong star-shaped sets (related to the dilations) and the definition of weak star-shaped sets (related to the horizontal line segments). We construct counterexamples showing that the two definitions are not equivalent. While the first definition is important for some PDE application, the second weaker definition is key for convexity. In fact weak star-shaped with respect to all interior points is equivalent to the horizontal convexity (and all other equivalent notions known in the Heisenberg group). Beside that we study the relation between many different notions of star-shaped sets and their relations with convexity and convex functions. These results have been applied in the study of geometrical properties for level sets of nonlinear subelliptic PDEs.
Zhuo Min Harold Lim (Cambridge, UK)
The Chern-Simons-Schrödinger system is a gauge-covariant version of the cubic nonlinear Schrödinger equation in two space dimensions. It describes the effective dynamics of a large system of nonrelativistic
charged quantum particles in the plane, interacting with each other and also with a self-generated long-range electromagnetic field. I will present my recent work establishing well-posedness in the energy space and scattering for the defocusing system, which describes a repulsive binary interaction. The scattering result is surprising from a physical point of view, and reflects subtle cancellations in the long-range electromagnetic interactions
Panagiota Birmpa (Sussex, UK)
We will discuss the power needed to force a motion of a interface between two different phases of a given ferromagnetic sample with a prescribed speed V. In this model, the interface is the non-homogeneous stationary solution of a non local evolution equation. Considering a stochastic microscopic system of Ising spins with Kac interaction evolving in time according to Glauber dynamics, we assign the cost functional which penalizes deviations from the solutions of the mesoscopic evolution equation by considering the underlying microscopic process. Then, we study the optimal way to displace the interface.
Oana Lang (Imperial, UK)
The aim of the talk/poster is to present a stochastic filtering problem consisting of a signal that models the motion of an incompressible fluid below a free surface whenthe vertical length scale is much smaller than the horizontal one. The evolution of the two-dimensional rotating system is represented by an infinite dimensional stochastic PDE and observed via a finite dimensional observation process. The deterministic
part of the SPDE consists of a classical shallow water equation (with an added viscosity term) and a new type of noise, namely the one introduced in [2]. Although this is a single layer model, therefore it does not completely reflect the complex stratification of the real atmosphere, it allows for important geophysical phenomena such as gravity and Rossby waves, eddy formation and geophysical turbulence.
References:
[1] A. Bain, D. Crisan, Fundamentals of Stochastic Filtering, Springer, 2009
[2] D. Holm , Variational Principles for Stochastic Fluid Dynamics, 2015
[3] G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, 2005