Abstract: Nonlocal repulsive interaction equations and nonlinear diffusion equations are popular choices to model repulsive behaviour in biomathematics and population dynamics. Due to their applicability to real-world scenarios, understanding the fine relationship between nonlocal dispersal and nonlinear diffusion has gained considerable traction in recent years. In this talk, we depart from a multiphase viscoelastic tissue growth model, in which the velocity obeys a Brinkman-type law, and investigate the “vanishing viscosity” localisation limit to derive a degenerate cross-diffusion system (where the velocity is given by Darcy’s law). Subsequently, we discuss the case of anisotropic diffusion and phenotypically stratified populations.

Abstract: Mean curvature flow is the gradient flow of the area functional where an embedded hypersurface evolves in direction of its mean curvature vector. This constitutes a natural geometric heat equation for hypersurfaces, which ideally will evolve the embedding into a nicer shape. But due to the nonlinear nature of the equation singularities are guaranteed to form. Nevertheless, a key observation in geometry and physics is that generic solutions, obtained by small perturbations, can exhibit simpler singularities. In this direction, a conjecture of Huisken posits that a generic mean curvature flow encounters only the simplest singularities. We will discuss work together with Chodosh, Choi and Mantoulidis which together with recent work of Bamler-Kleiner establishes this conjecture for embedded hypersurfaces in R^3.

Abstract: We consider a general compressible viscous, heat and magnetic conducting fluid described by compressible Navier–Stokes–Fourier system coupled with induction equation. In particular, we do not assume conservative boundary condition for temperature and allow heating or cooling on the surface of the domain We are interested in mathematical analysis when Mach, Froude, and Alvé numbers are small - converging to zero. We give a rigorous mathematical justification that i the limit, in case of low stratification, one obtains a modified Oberbeck–Boussinesq–MHD system with nonlocal term or non-local boundary condition for the temperature deviation. Choosing proper form of background magnetic field, gravitational potential and domain between parallel plates one also found that the flow is horizontal. The proof is based on the analysis of weak solutions to primitive system and relative entropy method. This is a recent joint work with Florian Oschmann and Piotr Gwiazda.

Abstract: I will discuss our advances on the analysis of the so-called dissipative Aw-Rascle model. This system of PDEs is known to describe the pedestrian flow, but in certain situations, it is also equivalent to the compressible Euler-alignment system. Various existence and uniqueness results will pre presented along with some singular limit results.