In this talk, we shall explain how transportation networks emerge as a self-regulating process, with a particular focus on applications in biology (leaf venation in plants, neuronal networks in animals). We start by introducing a purely diffusive model with tensor-valued diffusivity, derived as a gradient flow of a broad class of entropy dissipations. The introduction of a prescribed electric potential leads to the Fokker-Planck equation. We show that with quadratic entropy density modeling Joule heating, the model is convex with respect to the diffusivity tensor. Coupling with the Poisson equation for the electric potential we obtain the Poisson-Nernst-Planck system. The formal gradient flow of the associated entropy loss functional gives an evolution equation for the diffusivity tensor, coupled with two auxiliary elliptic PDEs. Finally, introducing a metabolic cost term, we obtain a system modeling network formation in the biological context.
- 1999-2003 MSc. in Applied Mathematics, Charles U. in Prague
- 2003-2008 PhD. in Applied Mathematics, U. of Vienna
- 2008-2009 Postdoc in PDE Analysis, Vienna U. of Technology
- 2009-2012 Postdoc in PDE Analysis, Johann Radon Institute, Linz
- 2012-2020 Research Scientist, CEMSE, KAUST
- 2021-present Senior Research Scientist, CEMSE, KAUST