Grain boundaries are interfaces across which crystal orientation changes. Traditional analysis suggest that grain boundary migration is effectively motion by mean curvature. However, this view is not in accordance with what we now know as the structure of grain boundaries on an atomic level. Just as surfaces of crystals move and roughen through the dynamics of surface steps, grain boundary dynamics is controlled by the motion of line defects known as disconnections. Unlike surface steps, disconnections are sources of long range stress (i.e., they have both dislocation and step character). In this talk, I will present an approach for understanding the motion of grain boundaries via disconnection motion and the relationship between disconnections and the underlying crystal structure. Next, I will discuss the homogenization of this type of disconnection-driven motion to yield a crystal-structure specific grain boundary equation of motion. I will then show several atomistic and numerical examples of “tame" GB motion (i.e., in bicyrstals) and GB motion “in the wild” (within polycrystals). This is very much a work in progress so I will also outline approaches for generalizations to general interface controlled microstructure evolution.
In this talk, we will focus on a kinetic equation modeling the spatial dynamics of a set of particles subject to intra-specific competition. This equation is motivated by the study of the propagation of biological populations, such as the Escherichia coli bacterium or the cane toad Rhinella marina, for which the classical diffusion approximation underestimates the actual range expansion of the species. We will use the optics geometrics approach as well as Hamilton-Jacobi equations to study spreading results for this equation. As we will see, the multi-dimensional case engenders technical difficulties, and possible over-representation of fast individuals at the edge of the front.
In this talk I present a general framework to model cell-cycle structured
populations living in a chemostat. The main examples are E. coli, or yeast, both
model organisms which have been intensively investigated to understand cell-cycle
controls. In this simple case the cells' cell cycle influence each other only by the level
of nutrients found in the culture medium. Otherwise the cell cycle in each cell behaves
autonomously. As the cell-cycle depends on many cell-internal biochemical concentrations,
most importantly on the cyclin protein family, the dynamical system describing the internal
cell dynamics can be of arbitrary high dimension, making the model extremely complex.
In order to investigate the model behaviour we decided not to use numerical time-integration,
but numerical continuation and bifurcation techniques. The respective numerical algorithm
is again of immense complexity, and uses a cell cohort discretisation. The plan is to refine
the model in future, most importantly bringing it to a tissue level in order to describe
I will talk about the fluid equations used to model pedestrian motion and traffic. I will present the compressible-incompressible Navier-Stokes two phase system describing the flow in the free and in the congested regimes, respectively. I will also show how to approximate such system by the compressible Navier-Stokes equations with singular pressure for the fixed barrier densities, together with some recent developments for the barrier densities varying in the space and time.
This is a talk based on several papers in collaboration with: D. Bresch, C. Perrin, P. Degond, P. Minkowski, and L. Navoret.