Francis Filbet. Université de Toulouse, France.  Personal website

Title: Numerical methods for kinetic models of self-organized dynamics

We deal with the numerical resolution of kinetic models for systems of self-propelled particles subject to alignment interaction and attraction-repulsion. We focus on the kinetic model  where alignment is taken into account in addition of an attraction-repulsion interaction potential. We apply a discontinuous Galerkin method for the free transport and non-local drift velocity together with a spectral method for the velocity variable. Then, we analyse consistency and stability of the semi-discrete scheme. We propose several numerical experiments which provide a solid validation of the method and its underlying concepts.

Ansgar Jüngel. Vienna University of Technology, Austria.  Personal website

Title: Cross-diffusion models for multispecies systems in biology

Abstract: The description of the dynamics of individuals for multiple species (e.g. populations, biological cells, ions) on the macroscopic level usually leads to strongly coupled parabolic systems. The diffusion matrix of these cross-diffusion systems is typically neither symmetric nor positive definite, which complicates the analysis. The idea is to reveal a gradient-flow or entropy structure, which allows for gradient estimates and in certain application for lower and upper bounds without applying a

maximum principle. In this talk, the entropy structure is detailed,

global existence results are given, and the exponential time decay to equilibrium is presented. The entropy technique is applied to models from population dynamics, tumor growth, and modified chemotaxis.

Giulio Schimperna. Università di Pavia, Italy.  Personal website

Title: On some mathematical models for tumor growth

Abstract: My talk will be devoted to discuss some mathematical aspects of diffuse interface models for tumor growth. These models usually involve a (possibly multi-component) Cahn-Hilliard equation describing the evolution of the various cell types, coupled with a Navier-Stokes (or Darcy, or Brinkman) law for the macroscopic flow velocity, and possibly with other evolutionary relations for additional significant variables (like for instance the concentration of some nutrient or drug). I will first present a survey about known results and then focus on some specific model focusing in particular on regularity and long-time behavior results.

Kristoffer G. van der Zee. School of Mathematical Sciences University of Nottingham, UK.  Personal website

Title: Phase-field tumour growth: Modelling, analysis and simulation

Abstract: In this lecture, I will consider nonlinear PDE systems relevant to the growth of cancerous tumours, which can be classified as phase-field models (also called diffuse-interface models). While some phase-field models are very classical, having their origins in material science (i.e., the Cahn-Hilliard equation and Allen-Cahn equation), they are an emerging paradigm for evolving-interface problems in the biological sciences. The key idea in phase-field models is that interfaces have a finite thickness (so-called diffuse interfaces), and are implicitly captured by the phase field, in contrast to sharp-interface models (e.g., moving-boundary problems).

  I will focus on an elementary phase-field tumour-growth model consisting of a higher-order, singularly-perturbed, semi-linear parabolic PDE coupled  to a reaction-diffusion PDE (for tumour and nutrient concentration, respectively). For this system, I will discuss its underlying gradient-flow structure, thermo-mechanical foundations, and sharp-interface limit. Numerical examples will be presented based on special energy-stable time-stepping schemes, which mimic the gradient-flow structure at the discrete level. Recent extensions to phase-field models involving the growth of blood vessels (tumour angiogenesis) will also be discussed.