EMERGENCE OF BIOLOGICAL TRANSPORTATION NETWORKS AS A SELF-REGULATED PROCESS

Jan Haskovec*, Peter Markowich, Simone Portaro

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

We study self-regulating processes modeling biological transportation networks. Firstly, we write the formal L2-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 L2-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.

Original languageEnglish (US)
Pages (from-to)1499-1515
Number of pages17
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume43
Issue number3-4
DOIs
StatePublished - Mar 2023

Bibliographical note

Publisher Copyright:
© 2023 American Institute of Mathematical Sciences. All rights reserved.

Keywords

  • biological network formation
  • convexity
  • Entropy dissipation
  • gradient flow
  • Poisson-Nernst-Planck

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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