Emergence of biological transportation networks as a self-regulated process
| Autor: | Haskovec, Jan, Markowich, Peter, Portaro, Simone |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Discrete and Continuous Dynamical Systems, 2023, 43(3&4): 1499-1515 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.3934/dcds.2022159 |
| Popis: | We study self-regulating processes modeling biological transportation networks. Firstly, we write the formal $L^2$-gradient flow for the symmetric tensor valued diffusivity $D$ of a broad class of entropy dissipations associated with a purely diffusive model. The introduction of a prescribed electric potential leads to the Fokker-Planck equation, for whose entropy dissipations we also investigate the formal $L^2$-gradient flow. We derive an integral formula for the second variation of the dissipation functional, proving convexity (in dependence of diffusivity tensor) for a quadratic entropy density modeling Joule heating. Finally, we couple in the Poisson equation for the electric potential obtaining the Poisson-Nernst-Planck system. The formal gradient flow of the associated entropy loss functional is derived, giving an evolution equation for $D$ coupled with two auxiliary elliptic PDEs. Comment: 16 pages |
| Databáze: | arXiv |
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