Self-regulated biological transportation structures with general entropy dissipation: 2D case and leaf-shaped domain
<|reference_start|>Self-regulated biological transportation structures with general entropy dissipation: 2D case and leaf-shaped domain: In recent years, the study of biological transportation networks has attracted significant interest, focusing on their self-regulating, demand-driven nature. This paper examines a mathematical model for these networks, featuring nonlinear elliptic equations for pressure and an auxiliary variable, and a reaction-diffusion parabolic equation for the conductivity tensor, introduced in \cite{portaro2022emergence}. The model, based on an energy functional with diffusive and metabolic terms, allows for various entropy generating functions, facilitating its application to different biological scenarios. We proved a local well-posedness result for the problem in H\"older spaces employing Schauder and semigroup theory. Then, after a suitable parameter reduction through scaling, we computed the numerical solution for the proposed system using a recently developed ghost nodal finite element method \cite{astuto2024nodal}. An interesting aspect emerges when the solution is very articulated and the branches occupy a wide region of the domain.<|reference_end|>
@article{astuto2024self-regulated,
    title={Self-regulated biological transportation structures with general entropy
  dissipation: 2D case and leaf-shaped domain},
    author={Clarissa Astuto and Peter Markowich and Simone Portaro and Giovanni
  Russo},
    journal={arXiv preprint arXiv:2408.15680},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.15680},
    primaryClass={math.AP cs.NA math.NA}
}