A proof of Kirchhoff's first law for hyperbolic conservation laws on networks

Alexandre M. Bayen; Alexander Keimer; Nils Müller; Alexandre M. Bayen; Alexander Keimer; Nils Müller

doi:10.3934/nhm.2023078

Networks and Heterogeneous Media

2023, Volume 18, Issue 4: 1799-1819. doi: 10.3934/nhm.2023078

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Research article

A proof of Kirchhoff's first law for hyperbolic conservation laws on networks

1.
Department of Electrical Engineering and Computer Sciences, Department of Civil and Environmental Engineering, University of California, Berkeley, United States of America
2.
Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
3.
Max Planck Institute for Software Systems, Saarland Informatics Campus, Germany

Received: 04 May 2023 Revised: 02 October 2023 Accepted: 16 October 2023 Published: 13 November 2023

In dynamical systems on networks, Kirchhoff's first law describes the local conservation of a quantity across edges. Predominantly, Kirchhoff's first law has been conceived as a phenomenological law of continuum physics. We establish its algebraic form as a property that is inherited from fundamental axioms of a network's geometry, instead of a law observed in physical nature. To this end, we extend calculus to networks, modeled as abstract metric spaces, and derive Kirchhoff's first law for hyperbolic conservation laws. In particular, our results show that hyperbolic conservation laws on networks can be stated without explicit Kirchhoff-type boundary conditions.
- networks,
- conservation laws,
- analysis of pdes,
- mathematical modeling
Citation: Alexandre M. Bayen, Alexander Keimer, Nils Müller. A proof of Kirchhoff's first law for hyperbolic conservation laws on networks[J]. Networks and Heterogeneous Media, 2023, 18(4): 1799-1819. doi: 10.3934/nhm.2023078

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Abstract

In dynamical systems on networks, Kirchhoff's first law describes the local conservation of a quantity across edges. Predominantly, Kirchhoff's first law has been conceived as a phenomenological law of continuum physics. We establish its algebraic form as a property that is inherited from fundamental axioms of a network's geometry, instead of a law observed in physical nature. To this end, we extend calculus to networks, modeled as abstract metric spaces, and derive Kirchhoff's first law for hyperbolic conservation laws. In particular, our results show that hyperbolic conservation laws on networks can be stated without explicit Kirchhoff-type boundary conditions.

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