## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 1499-1515 |

Number of pages | 17 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 43 |

Issue number | 3-4 |

DOIs | |

State | Published - 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