PDE models for transportation networks (Peter A. Markowich)
[P4, Markowich] focuses on a novel class of PDE-based transportation network models. These can be used to describe, for example, the formation, evolution, adaptation and flow dynamics of networks such as leaf venation in plants, or neuron networks and blood capillary networks in animals. By relying on first principles, transportation network models are intrinsically different from classical descriptive deterministic or random graph-based models. They consist of a Poisson equation for the driving force (e.g. the pressure of the transported fluid or the electric field generated by the ions transported in the network) with a pressure diffusivity tensor which depends on the conductance vector or tensor, coupled nonlinearly to a reaction-diffusion system for the network conductance.
The models can typically be obtained as formal gradient flows of highly non-convex network energy functionals. We shall study basic PDE analysis questions like existence of weak and strong solutions, uniqueness issues as well as regularity and stability of solutions, also in cooperation with [P9, Schmeiser] and [P11, Stefanelli]. Challenges stem from the occurrence of thin substrate layers with large conductance gradients and ramifications in certain scaling limits giving singular measure transient solutions as well as stationary states. Numerical approaches taking into account the particular solution structure shall also be studied in cooperation with [P7, Perugia] and [P10, Schöberl].