Speaker
Description
We present a Young measure approach to the convergence analysis of a fully discrete finite-volume scheme in multiple space dimensions for a class of cross-diffusion systems with a rank-deficient diffusion matrix. Such systems arise in the modelling of biological multi-component materials with incomplete diffusive mixing and may exhibit fully or partially segregated steady states.
The main analytical difficulty stems from the strongly coupled hyperbolic--parabolic structure of the system, together with a porous-medium-type degeneracy. Our analysis relies on the existence of two dissipated functionals, a Rao-type energy and the Boltzmann entropy, and on a discretisation preserving these structures. To pass to the continuum limit, we devise a Young measure framework, inspired by related developments in fluid dynamics, and introduce a vanishing artificial diffusion to identify the limiting fluxes. A weak--strong uniqueness principle then upgrades weak measure-valued convergence to strong convergence whenever the continuum model admits a strong solution.
This is joint work with Ansgar Jüngel (TU Wien).