Workshop on PDEs: Modelling,
Analysis and Numerical Simulation


Granada, January 8-16


  Wednesday 8 Thursday 9 Friday 10
09:30-11:30 Gero Friesecke Sylvia Serfaty Eva Löcherbach
    Poster session  
12:00-14:00 Gero Friesecke Ulrik Fjordholm Eva Löcherbach
    Poster session  


Sylvia Serfaty Ulrik Fjordholm  



Computing measure-valued and statistical solutions of hyperbolic conservation laws
Ulrik Fjordholm

There is an increasing body of evidence that common models of inviscid fluids, such as the (in)compressible Euler equations, are ill-posed in the sense that they admit infinitely many entropy solutions for the same initial data. The concept of measure-valued and statistical solutions suggests tackling this issue by viewing the solution as a probability distribution over all possible entropy solutions. In this mini-course I aim to present some of the tools for working with statistical solutions, as well as practical algorithms for computing such solutions.

Crystallization in classical particle systems.
Gero Friesecke

The emergence of crystalline order is ubiquitous in physics, biology, and mathematics: atoms self-assemble into three-dimensional crystals; messy-looking proteins assemble into perfectly regular two-dimensional virus shells; densest sphere packings appear to be crystalline. Related effects appear in continuum models (vortex lattices in nonlinear Schroedinger equations; periodic Voronoi tesselations in optimal transport). Mathematically, these phenomena are subtle. Previous understanding for 2D classical particle systems by pioneers like Heitmann, Radin, Theil, Mainini, Stefanelli relies on highly technical, puzzlingly detailed, model-specific estimates. After a general introduction, I present a new viewpoint inspired by continuum mechanics which rigorously and exactly decomposes discrete 2D many-particle energies into elastic, surface, defect, and topological contributions, via methods from discrete differential geometry (used for the first time in crystallization problems). This approach, developed jointly with Lucia De Luca (Rome), yields a much simpler proof of the Heitmann-Radin crystallization theorem, and appears to be very promising for future research.

Probabilistic models for networks of spiking neurons
Eva Löcherbach

We will discuss a class of recently introduced models proposing to describe networks of neurons as stochastic processes with memory of variable length. These are non- Markovian processes in high or infinite dimension in which the past dependence of transition probabilities or intensities has a range that is finite but depends on the particular history. Starting from existence results, we will briefly study related mean-field models in continuous time and their large population limits. We will also discuss the relation with associated Piecewise Deterministic Markov Processes (PDMP’s) and state results concerning their longtime behavior. Finally, we will touch an important problem of statistical inference in such models : the estimation of the of the neuronal interaction graph.

My lectures will be based on joint work with Antonio Galves, Aline Duarte and Guilherme Ost.

Mean-field limits for Coulomb-type dynamics
Sylvia Serfaty

Systems of N particles evolving according to the gradient flow (or similar conservative flow) to some interaction energy typically converge as N gets large to mean field limit evolution PDEs, this has been proven in the case of sufficiently regular interactions. The case of Coulomb or more singular interactions had remained an open question. We will describe a result based on a new modulated energy approach which allows to treat Coulomb and Riesz interaction (inverse power s of the distance with s between d-2 and d where d denotes the dimension).

Time allowing we also discuss related results for Ginzburg-Landau vortex dynamics.



School of Sciences, Technology and Engineering