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Continuum Modeling of Transportation Networks with Differential Equations

Abstract: Motivated by recent papers describing rules for biological network formation in discrete settings, we propose  an elliptic-parabolic system of partial differential equations. The model describes the pressure field due to a Darcy type equation and the dynamics of the conductance vector under pressure force effects with a diffusion rate representing randomness in the material structure. After a short overview of the principles of discrete network modeling, we show how to derive the corresponding macroscopic (continuum) description. The highly unusual structure of the resulting PDE system induces several interesting challenges for its mathematical analysis. We give a short overview of the analytical tools that can be used to overcome them. In particular, we present results regarding the existence of weak transient solutions of the system and we study the structure and stability properties of steady states that play a central role to understand the pattern formation capacity of the system. We present results of systematic numerical simulations of the system that provide further insights into the properties of the network-type solutions.