Self-regulated biological transportation structures with general entropy dissipation: 2D case and leaf-shaped domain
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910580873363456 |
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| author | Astuto, Clarissa Markowich, Peter Portaro, Simone Russo, Giovanni |
| author_facet | Astuto, Clarissa Markowich, Peter Portaro, Simone Russo, Giovanni |
| contents | 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 Hölder 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_15680 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Self-regulated biological transportation structures with general entropy dissipation: 2D case and leaf-shaped domain Astuto, Clarissa Markowich, Peter Portaro, Simone Russo, Giovanni Analysis of PDEs Numerical Analysis 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 Hölder 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. |
| title | Self-regulated biological transportation structures with general entropy dissipation: 2D case and leaf-shaped domain |
| topic | Analysis of PDEs Numerical Analysis |
| url | https://arxiv.org/abs/2408.15680 |